Module in Math i

64
Fir st Deg ree Equ

Transcript of Module in Math i

Page 1: Module in Math i

First

Degree Equatio

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TOPIC: Introduction to First Degree Equations and Inequalities in One Variable

OBJECTIVES:

1. Distinguish between mathematical phrases and sentences2. Distinguish between expressions and equations3. Distinguish between equations and inequalities

WARM UP:

a. Encircle the expression that expresses a complete thought.1. x + 42. 4x = 123. 10 > x4. 5x – 85. 4y ≠ 0

b. Box the equation and encircle the inequality.1. 3x + 4 ≤ 102. 7 – x = 03. 12x – 4 ≠ 204. 12 > x5. 3y = 12

A mathematical sentence is a group of mathematical symbols which expresses a complete mathematical thought. A mathematical sentence whether numerical or algebraic, expresses either an equality or inequality between numbers. Equations 2x + 4 = 10, x > 4 and 4 < 10 are examples of mathematical sentences. On the other hand, 2x + 4, x – 10 and 4x are examples of mathematical phrases.

2x + 4 = 10, 4 = -x +3, and 7m – 1 = 3m + 5 are examples of equations in one variable. Equations are mathematical sentences which use the equality sign (=). It is a statement of equality of two algebraic expressions which have the same numerical value.

Inequalities are mathematical sentences which use the relation symbols ≠, < or > and ≤ or ≥. Examples of these are 2 > 3x + 1, 5x - 10x ≠ x + 1 , 1 – y < 0, 5x > 2 and x < -3. They are mathematical statements which state that two qualities are not equal.

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EXERCISES

A. Read each expression carefully. Encircle P if it is a mathematical phrase and encircle S if it is a mathematical sentence.

P S 1. 12 - x P S 6. 7x – 9y + z

P S 2. 4y = 24 P S 7. 6y – 4 =25

P S 3. 2(x + 5) P S 8. 4x <

P S 4. 5x > 10 P S 9. 7y + 5 ≤ 15

P S 5. 3x = 0 P S 10. 3x- 5y

B. Write E if the mathematical sentence is an equation and I if it is an inequality.

______ 1. 4 – y ≤ 10 ______ 6. 3 (2x+ 6) = 15

______ 2. 5 + 7x = 35 ______ 7. 4x ≠ x - 6

______ 3. 4y > 12 ______ 8. x = 5x - 12

______ 4. 4 ≥ 2x ______ 9. 7y + 4 = 15

______ 5. 3x + 2 = 4x - 6 ______ 10. 6 < y <2

C. Give your own examples of the following:

Mathematical Sentence Mathematical Phrase1. ___________________ 1. ___________________2. ___________________ 2. ___________________3. ___________________ 3. ___________________4. ___________________ 4. ___________________5. ___________________ 5. ___________________

Equation Inequality

1. ___________________ 1. ___________________2. ___________________ 2. ___________________3. ___________________ 3. ___________________4. ___________________ 4. ___________________5. ___________________ 5. ___________________

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TOPIC: Translation of Verbal Statements into Equations and Inequalities

OBJECTIVE:

Translate verbal statements into mathematical sentences and vice-versa

EXERCISES

A. Translate the following verbal expressions into mathematical phrases or sentences. Use x to represent the unknown and write your answers in the space provided at the right.

1. The product of 18 and z ________________2. Thrice the difference of x and y ________________3. A number is at most 7. ________________4. A number is at least 25 . ________________5. x subtracted from 9 equals -15. ________________6. Twice a number decreased by 4 is 11. ________________7. x is decreased by two times y. ________________8. Twice the sum of a number and 9 ________________9. One-third the difference of x and y ________________10. Thirteen more than five times a number ________________11. Four less than the quotient of a and b ________________12. A number divided by 20 ________________13. Twice a number diminished by 24 ________________

14. Nine multiplied by the sum of a number and three is ________________equal to 48.

Observe how the equality or inequality symbols are used to translate the following verbal statements into mathematical sentences.

1. The sum of a number and ten is fifteen. x + 10 = 152. 5y is greater than or equal to 40. 5y ≥ 403. 6 is increased by 2 times x is equal to 24. 6 + 2x = 244. The difference when 4 is subtracted from n is greater than 12. n – 4 > 125. The product when 4 is multiplied by the sum of m and 1 is less 4(m + 1) ≤ 20

than or equal to 20.6. x less 5 is 15. x – 5 = 157. x less than 5. X < 58. 4x diminished by 10 is 20 4x – 10 = 20

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15. Four less than three times a number ________________B. Translate the following mathematical phrases/statement into verbal phrases/sentences.

1. x ≥ 8 ________________________________________________

2. 12x ________________________________________________3. 4x - 6 ________________________________________________

4. 18 ÷ (5 - x )________________________________________________

5. y – 6 = 10 ________________________________________________

6. x + 12 > 20 ________________________________________________

7. 5 < x +4 ________________________________________________

8. 7x < 12 ________________________________________________

9. 10K + 12 ________________________________________________

10. 3x + 2 = 11 ________________________________________________

C. RIDDLE

4 2 3 5 1 7 6 5 1

Translate the following verbal sentences into mathematical phrase. Use the DECODER to answer the riddle.1. Three times x more than 4 equals 12.2. Three times x is greater than 4 plus 12.3. Three times x less than 4 equals 12.4. Three times x is less than 4 plus 12.5. Three times the product of x and 4 equals 12.6. Three times the difference of x and 4 equals 12.7. Three times the sum of x and 4 equals 12.

What do you call the ordinal order that is not correct?

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D E C O D E R

G3x + 4 = 12

R3x > 4 + 12

O3x – 4 = 12

W3x < 4 + 12

N3(4x) = 12

I3(x – 4) = 12

K3(x + 4) = 12

E3(4 ÷x) = 12

TOPIC: Solution Set of a First- Degree Equation or Inequality in One Variable From a Replacement Set

OBJECTIVES:

1. Define the solution set of a first degree equation or inequality2. Find the solution set of simple equations and inequalities in one variable from a given

replacement set

Sentences which contain variables are called open number sentences, or simply open sentences. An open sentence generally becomes a statement which is either true or false when we give a value to the variable. The set of numbers which makes the open sentence true is called the solution set or root of the sentence.

Sometimes the set from which the solution set is to be chosen is specified. We then call this the domain of the variable or replacement set. Study the following examples.

1. x - 6 = 4 Domain: ( 7, 8, 9, 10)Solution Set: (10)

X = 10

Substituting each of the elements in the domain in the given equations results in one true sentence.

2. 4x – 5 = 45 Domain: set of counting numbersSolution Set: (12)

x = 12

By substitution, 12 satisfies the given equation.

3. 2x + 1 < 7 Domain: ( 3, 2, 1 0 , -1, -2 ,-3, -4, …)

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Solution Set: (2, 1, 0, -1, -2, -3, -4, …)

In this example, the solution set has finite number of elements. Substitute each element in the solution set to test whether each satisfies the given open sentence.

EXERCISES

A. Use (-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5) as your replacement set to find the solution of the following:

1. x > 0 ____________________2. x ≥ 4 ____________________3. x + 2 = -1 ____________________4. x – 3 = 5 ____________________5. 2x – 1 = 1 ____________________6. 2x + 5 > 1 ____________________7. 3x – 2 < 4 ____________________8. x + 2 ≥ 3x – 2 ____________________9. 6x – 5 ≤ 2x + 7 ____________________10. 8x + 3 ≥ 5x + 3 ____________________

B. Replace the variable by each number in the given replacement set and tell whether the resulting sentence is true.

1. x + 7 = 4 (3, -3) _____________ _____________2. 3y = 18 (6, 9) _____________ _____________3. 13 = x – 2 (11, 15) _____________ _____________4. 2x + 2 > 3 (0, 1) _____________ _____________5. 4x – 1 < 3 (2, -2) _____________ _____________6. 3x – 2 = 7 (3, 4) _____________ _____________7. 3x – 8 = 2x – 7 (1, -15) _____________ _____________8. 4x – 2 = 2x +8 (5, 2) _____________ _____________9. x + 2 = 3 (12, 10) _____________ _____________

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410. 2x -1 = 5 (15, 8) _____________ _____________

3

C. RIDDLE

_____ _____ _____ _____ 1 2 3 4

_____ _____ _____ _____ _____ _____ _____ _____ 5 6 7 8 9 10 11 12

The replacement set for the variable in each of the following sentences is {-3, 0, 1, 2, 3, 4, 5, 6, 7, 9}. Find the solution or root of each sentence.

1. Y + 4 > 9 = ___________ 7. X + 9 = 4 = ___________2. X + 2 = 7 = ___________ 8. w/3 = 2 = ___________3. 4m = 12 = ___________ 9. X – 1 = -4 = ___________4. 3R = 6 = ___________ 10. N + 3 ≤ 5 = ___________5. X + 1 < 6 = ___________ 11. 9x – 3 = 6 = ___________6. X – 2 > 5 = ___________ 12. 2m – 4 = 4 = ___________

L(4, 3, 2, 1, 0, -3)

B(-3, 6)

E-3

T5

Y(7, -3)

O9

I(6, 7, 9)

P4

U1

D(2, 1, 0, -3)

X(4, 7)

CNo solution

I3

K6

S2

W(1,9)

What is the best way to know if a piece of jewelry is expensive?

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TOPIC: Solution Set of Equations and Inequalities in One Variable on a Number Line

OBJECTIVE:

1. Graph the solutions of equations and inequalities on a number line

At this point, let us consider a point M on the number line whose coordinate is m. Take note of the graph of the relation described in each of the following cases: (Here, x represents the solution.)

1. The coordinates of all points to the right of M are greater than m and all points to the left of M have coordinates less than m.

In symbols, x = m

2.

The shaded potion of the graph shows all points whose coordinates are less than m, that is, x < m. The small unshaded circle on top of m means that m is not included.

3.

Here, the shaded potion of the graph represents all numbers greater than m, In symbols, x > m.

M●m

M○m

M○m

M●m

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4.

The shaded circle on top of m and the arrow from m to the left on the number line indicates all numbers to the left of m, including m. In symbols, x ≤ m.

5.

The graph above shows all numbers to the right of m, including m. In symbols, x ≥ m.

EXERCISES

A. Graph the solution set of the following equalities and inequalities.

1. x = 6

2. x > 4

3. x < 6

4. x ≤ 6

5. x ≥ -7

6. x < -4

7. x ≥ 15

M●m

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8. x > -12

9. x ≤ 5

10. x ≥ 26

TOPIC: Basic Properties of Real Numbers

OBJECTIVE:

State / name the basic properties of real numbers

PROPERTIES OF REAL NUMBERS

Let a, b and c represent any real numbers (a, b, and c ϵ IR)

Addition MultiplicationCommutative

AssociativeIdentityInverse

a + b = b + a(a + b) + c = a + (b + c)

a+ 0 = 0 + aa + (-a) = (-a) + a

a•b = b •a(a•b)•c = a•(b•c)

a•1 = 1•aIf a is not zero, then a•1/a = 1 = 1/a•a

Distributive Property of Multiplication over AdditionA(b + c) = ab + ac and (b + c)a = ba + ca

EXERCISES

A. Name the property that justifies each of the following statements.1. 5 • 1 = 5 _______________________________________________2. 7 + (-7) = 0 _______________________________________________3. 6 + 0 = 6 _______________________________________________4. (3 + 2)5 = 3•5 + 2•5 _______________________________________________

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5. X + y = y + x _______________________________________________6. 5 + (x + y) = (5 + x) + y _______________________________________________7. 2a + 2b = 2(a + b) _______________________________________________8. (x + y) + z = x + (y + z) _______________________________________________9. ½ •2 = 1 _______________________________________________10. Y + 5 = 5 + y _______________________________________________

B. Use the indicated property to write an expression that is equivalent to each of the following expressions

1. 8 •7 (commutative property) ________________________2. 9 + 6 (commutative property) ________________________3. 3(5 +4) (distributive property) ________________________4. (8•4) + (5 + 7) (commutative property) ________________________5. (5 + x) + 3 (associative property) ________________________6. (8 + x)4 (distributive property) ________________________7. (7•5)6 (associative property) ________________________8. X + (2 + 14) (commutative property) ________________________9. 4•7 + 4•6 (distributive property) ________________________10. [(m + p) + r] + s (associative property) ________________________

C. SHARPEN YOUR SKILL

Complete the equations below to illustrate exactly one of the properties. Give the name of the property used.

1. 3 (5 + 7) = ________ + ____ ____ __________________________

2. 9 + ________ = 9 __________________________

3. 15 + ________ = 0 __________________________

4. 8 • ________ = 1 __________________________

5. 15 + 7 = ________ + ________ __________________________

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6. ________ • 8 = 9 • ________ __________________________

7. (3 •6 ) •4 = ________•(________•________) __________________________

8. 8 + ________ = 12 + ________ __________________________

TOPIC: Properties of Equalities

OBJECTIVE:

State and illustrate the different properties of equality.

PROPERTIES OF EQUALITY

For any numbers a, b, cReflexive : a = aSymmetric : If a = b, then b = a.Transitive : If a = b, and b = c then a = c.Substitution : If a = b, then a may be replaced by b.Addition : If a =b, then a + c = b + cSubtraction : If a =b, then a - c = b - cMultiplication : If a =b, then a •c = b • cDivision : If a =b and if c is not zero, a/c = b/c

EXERCISES

A. Apply the properties of equality on the following:1. 5 = 5 , add 4 __________________________________2. 9 = 9 , subtract 7 __________________________________3. 6 = 6 , multiply 6 __________________________________4. 10 = 10 , add 12 __________________________________5. 4 = 4 , divided by 20 __________________________________6. 18 = 18 , subtract 17 __________________________________7. 95 = 95 , add 45 __________________________________8. 80 = 80 , multiply by 5 __________________________________9. 90 = 90 , divide by 10 __________________________________10. 35 = 35 , subtract by 15 __________________________________

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B. Fill in the blank with the correct expression to make the statement true.1. 18 = _____________ Reflexive Property2. If 5 + 3 = 8 then 8 = ______________ Symmetric Property3. If 9 + 6 = 15 then 9 + 6 + 4 = 15 + ____________________ Addition Property4. If2 + 3 = 5 and 5 = 4 + 1 then _____________________ Transitive Property5. If 4 •5 = 20 then ___________________ = 4 • 5 Symmetric Property6. If 10 + 3 = 13 then 10 +3 - __________ = 13 – 5 Subtraction Property7. If 2 •3 = 6 •1 then (2 •3)5 = (6 •1) ____________ Multiplication Property8. If 5 •6 = 3 • 10 then 5 • 6 = 3 • 10 Division Property

2 ___9. ______________ = -1/4 Reflexive Property10. If ______________ = 9 then 9 = 5 + 4 Symmetric Property11.

C. RIDDLE

Cross out the boxes that illustrate the properties of equality. The numbers in the remaining boxes contain the answer.

_____ _____ _____ _____ _____ _____ mph

62 + 3 = 2 + 9-3)

13 + 5 = 3 + 5

02(-1/2) = 2(1/2)

56 + (-2) = 6 + (-2)

46 ÷ 5 = 6 • 5

7½(2) = ½(2)

06 ÷(6) = 6 ÷ (6)

83 + 6 = 3 + 9

97 • 9 = 7 ÷ 9

03 – (-2) = 3 – (-2)

38 + 3 = 8 + (-3)

0(6 + 5)2 = (6 + 5) 2

TOPIC: Solving First-Degree Equations in One Variable

OBJECTIVES:

1. Solve first degree equations in one variable

What is the speed of the fastest space craft?

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2. Apply the properties of equality in solving first-degree equations involving one variable

Warm-up:

A. Fill in the blanks with the appropriate integers.

1. ___ + 4 = 5

2. ___ - 6 = 2

3. ___ + 6 = -2

4. 7 - ___ = 10

5. 9 + ___ = 5

B. Warm-up: Solve for the variable.

6. 8k = 24

7. 33 = -3b

8. -4p = -36

9.

d5=7

10.

−x2

=5

PROPERTIES OF EQUALITIES1. Addition Property of Equality (APE)

Given the numbers a, b and c, if a = b, then a + c = b + c.2. Multiplication Property of Equality (MPE)

Given the numbers a, b and c, if a = b, then ac = b c.3. The Equivalence Relation

Given the numbers a, b, and cReflexive Property a = aSymmetric Property if a = b then b = aTransitive Property if a = b and b = c, then a = c.

The solution of a simple equation is facilitated by the use of certain properties of equalities. Study the examples below.

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1. Solve for x in the equation x + 2 = 8Solution : x + 2 = 8 Given equation

x + 2 + (-2) = 8 + (-2) Add -2 to both sides, or APEx +[ 2 + (-2)] = 8 + (-2) Associative Propertyx + [2 + (-2)] = 8 – 2 Definition of Subtractionx + 0 = 6 Additive inversex = 6 Additive Identity

2. Solve for x in the equation 2x = 16Solution : 2x = 16 Given equation

½(2x) = ½ (16) Multiply both sides by 1/2, or MPE

(1/2 •2)x = 8 Associative Property

1 • x = 8 Multiplicative Inversex = 8 Multiplicative Identity

3. Solve for x in the equation x – 6 = 9Solution : x – 6 = 9 Given equation

x – 6 + 6 = 9 + 6 Add 6 to both sides, or APEx = 15 Simplify

4. Solve for x in the equation x = 3 2

Solution: 2 • x = 3 • 2 Multiply both sides of the equation by 2 2

x= 6 Simplify

EXERCISES

A. Solve the following equations and check the solutions.1. x – 13 = 82. x -15 = 253. x – 9 = 374. x – 12 = 435. x – 16 = 526. x + 23 = 847. x + 14 = 938. x + 19 = 57

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9. x + 29 = -2910. x + 27 = 75

B. Solve for the value of each variable in set of real numbers.1. 9x = 1082. 3y = 453. 7m = 564. 3x = 1145. 8t = 1446. x/4 = 67. y/3 = 98. m/6 = -79. 3x/4 = 1210. 2y/3 = 16

TOPIC: Solving Other types of First Degree Equations

OBJECTIVES:

1. Solve equations using the properties of equality2. Solve equations containing parentheses

An equation is said to have been solved if the value of the unknown quantity is such that when substituted for it, the two members of the equation will be equal. The following are other types of equations in one variable that are solved using the properties of real numbers and properties of equality.

EXAMPLES

1. Solve the equation 5x – 13 = 37Solution: 5x – 13 = 37

5x – 13 + 13 = 37 + 13 Addition Property of equality5x = 505x = 50 Division Property of equality

5 5 X = 10

Check: 5(10) – 13 = 3750 -13 = 3737 = 37

2. Solve: 4x -8 = x + 19Solution: 4x -8 = x + 19

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4x -8 + 8 = x + 19 + 8 Addition Property of Equality4x = x + 274x – x = x – x + 27 Subtraction Property of Equality3x = 273x = 27 Division Property of equality

3 3X = 9

Check: 4 (9) – 8 = 9 + 1936 – 8 = 2828 = 28

3. Solve 2(x – 4) = 10 + 5xSolution: 2(x – 4) = 10 + 5x

2x – 8 = 10 + 5x DPMA2x – 8 + 8 = 10 + 8 + 5x Addition Property of Equality2x = 18 + 5x2x – 5x = 18 + 5x – 5x Subtraction Property of Equality-3x = 18-3x = 18 Division Property of equality

-3 -3x = -6

Check: 2 (-6 – 4) = 10 + 5 (-6)2(-10) = 10 – 30-20 = -20

EXERCISES:A. Solve for the following equations. Check your answer.

1. 4x + 3 = 272. 4x – 8 = 23. 2x + 3 = 74. 5x – 4 = -145. -3x + 7 = -14

B. Solve for the value of the variables and check.1. 9x = 15 + 4x2. 5x – 3 = 7 + 4x3. 5 ( x + 2) – 3(x + 3) = 154. 7x – 3 = 5x + 9

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5. 4x – 5 = 11 8

TOPIC: Solving First Degree Inequalities in One Variable by Applying the Properties of Inequality

OBJECTIVES:

1. State the properties of inequalities2. Determine the solution set of first degree inequalities in one variable by applying the

properties of inequality

EXAMPLES:

1. Solve the inequality x – 2 > 6Solution: x – 2 > 6

PROPERTIES OF INEQUALITIES

1. Addition Property of Inequality (API)Given the numbers a, b and c.If a < b, then a +c < b + c; if a > b, then a +c > b + c

2. Multiplication Property of Inequality (MPI)Given the numbers a, b and c.If a > b, then ac > bc, for every positive number c.If a > b, then a c < bc, for every negative number c.If a < b, then ac < bc for every positive number c.If a < b, then ac > bc, for every negative number c.

3. Trichotomy PropertyGiven the number x.If x = 0, then x is not positive and x is not negative.If x > 0, then x = 0 and x is not negative.If x < 0, then x = 0 and x is not positive.

4. Transitive Property of InequalityGiven the numbers a, b, and c.If a < b and b <c, the a < c.

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x – 2 + 2 > 6 + 2 add 2 to both sides of the inequalityx > 8

2. Solve the inequality -3x < 12Solution: -3x < 12

-1 (-3x) > -1 (12) MPI3 3x> -4

3. Solve the inequality x ≥ 7 6Solution: x ≥ 7 6

6 x ≥ 6 (7) multiply both sides of the inequality by 6 6X ≥ 42

4. Solve the inequality 3 – 7x ≤ 4x – 303 – 7x ≤ 4x – 30 Given3 + (– 7x) ≤ 4x + (– 30) Definition of Subtraction3 + (– 7x + -4x) ≤ 4x + (-4x) + (– 30) API3 + (– 11x) ≤ 0 + (– 30) Additive Inverse3 +(-3) + (– 11x) ≤ (– 30) + (-3) Additive Inverse0 + (– 11x) ≤ (– 33) Additive Inverse(-11x) ≤ (-33) Additive Identity-1 (-11x) ≥ -1 (-33) MPI11 11X ≥ 3 Multiplicative Identity

EXERCISES

A. Solve each inequality.1. 5x > 302. x + 5 < 73. x + 7 < 14. x ≤ 4

85. -8 + x > 4

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B. Solve the inequality.1. 7x + 3 < 6x + 52. 5x – 3 < 7

23. 2x + 5 ≥ x – 14. 10 – 2x ≤ x -25. 2(x – 5) > 3 + x

C. CHALLENGE1. The sum of two consecutive integers is less than 35. What is the greatest

possible pair?2. The difference between two integers is less than 96. The larger number is

245. Find the smallest value of the other integer.3. What number should be added to 11 to get a sum of at least 26?4. Find two consecutive positive odd numbers whose sum is at most 60.5. Find three positive integers such that 12 greater than five times the integer is

greater than 7.

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Linear Equations

in Two Variables

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TOPIC: Cartesian Coordinate Plane

OBJECTIVE:

1. Describe the Cartesian Coordinate Plane,2. Given a point on the coordinate plane, give its coordinate.

A coordinate plane is a plane figure consists of two perpendicular number lines. The horizontal number line is called the x-axis. The vertical number line is called the y-axis. The intersection point of two number lines s called the origin at (0,0).

Each point on the coordinate plane has two coordinates (x,y). The first number in the ordered pair is the x-coordinate or the abscissa. The second number is the y-coordinate or the ordinate.

We can count the number of units a point is to the left or right of the y-axis and the number of units up or down from the x-axis to find the coordinates of a point.

EXAMPLE:

Using the graph below, find the coordinates of the following points.

1. M2. A

1

Linear Equations

in Two Variables

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3. T4. H

1. Point M on the graph has coordinates (3,3) which means it is located 3 units to the right of the y-axis and 3 units above the x-axis.

2. Point A on the graph has coordinates (-3,4) which means it is located 3 units to the left of the y-axis and 4 units above the x-axis.

3. Point T on the graph has coordinates (-3,-3) which means it is located 3 units to the left of the y-axis and 3 units below the x-axis.

4. Point H on the graph has coordinates (3,-4) which means it is located 3 units to the right of the y-axis and 4 units below the x-axis.

EXERCISES

Give the coordinates of the following points.

1. Point A ________2. Point B ________3. Point C ________4. Point D ________5. Point E ________6. Point F ________7. Point G ________8. Point H ________9. Point I ________10. Point J ________

1

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TOPIC 5.2 Plotting of Points

OBJECTIVE:Given a pair of coordinates, plot the points.

If points in a coordinate plane can be named, points can also be plotted in the plane given their coordinates. In an ordered pair (x,y), the x-coordinate tells how far a point is to the right or left of the origin and the y-coordinate tells how far a point is above or below the origin.

To plot a given point given its coordinates,1. From the origin, count the number of x-units.

go to the right if the abscissa is (+). go to the left if the abscissa is (-).

2. From this point on the x-axis, count the number of y-units. go upward if the ordinate is (+). Go downward if the ordinate is (-).

EXAMPLEGraph the following points in a coordinate plane.

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1. M(5,6)2. A(-6,-4)3. T(4,-5)4. H(-4,3)

SOLUTION

1. For M(5,6) Start at the origin. Move 5 units to the right, then move 6 units

up.

2. For A(-6,-4) Start at the origin. Move 6 units to the left, then move 4

units down.

3. For T(4,-5) Start at the origin. Move 4 units to the right, then move 5 units

down.

4. For H(-4,3) Start at the origin. Move 4 units to the left, then move 3 units up.

EXERCISES

1. Point A(6,4)

2. Point B(3,6)

3. Point C(-5,-4)

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4. Point D(-5,2)

5. Point E(1,-6)

6. Point F(5,3)

7. Point G(-6,-2)

8. Point H(3,-5)

9. Point I(-4,-4)

10. Point J(-1,6)

TOPIC 5.2.1 Points in a Quadrant

OBJECTIVE:

Given the coordinates of a point, determine the quadrant s or axes where the point is located.

The coordinate plane is divided into four quadrants. The quadrants are numbered counterclockwise starting at the upper right.

The axes are not a part of any quadrant. A point on the x-axis or y-axis is not in a quadrant since it is on the boundary between quadrants.

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EXAMPLES:

In which quadrant is each point located?

1. A (3,-12) answer: Quadrant IV

2. B (8,6) answer: Quadrant I

3. C (-3,-15) answer: Quadrant III

4. D (-11, 10) answer: Quadrant II

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EXERCISES

Without plotting the points, determine the quadrants or axes which each point can be found.

1. (4,7) __________________

2. (8,-2) __________________

3. (10,-1) __________________

4. (-15,9) __________________

5. x= -8 __________________

6. (-7,6) __________________

7. (-9,-5) __________________

8. y=12 __________________

9. (-3,-3) __________________

10. x= 0 __________________

TOPIC 5.3 Linear Equations in Two Variables

OBJECTIVES

1. Define a linear equation in two variables.2. Construct a table of values for x and y given a linear equation in two variables.

A linear equation in two variables has the standard form

Ax +By = C

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where A,B and C are real numbers and both A and B are not equal to zero. The exponent on each variable must be equal to 1. Also, we cannot have a term which contains the product or quotient of the two variables.

EXAMPLES:

1. 3x + y =2 a linear equation in two variables in standard form2. y= x +1 a linear equation in two variables not in standard form3. x2 + y = 7 not a linear equation in two variables4. xy + x = -3 not a linear equation in two variables

Suppose we are asked to construct a table of the linear equation y= x+1, we can choose convenient values of x or a replacement set.

Let x = 0, 1 and 2 Solving for y, we have :

If x=0, y= 0+1 = 1 therefore, (0,1)

If x=1, y=1+1 =2 therefore, (1,2)

If x=2, y=2+1 =3 therefore,(2,3)

So, the table of values for x and y are

x 0 1 2y 1 2 3

EXERCISES

A. Complete the ordered pairs using the given equations.

1. y= x + 10 (0,__), (__,0), (-2,__)

2. x= 2y + 1 (0,__), (__,0), (3,__)

3. y + 5 = x (0,__), (__,0), (6,__)

4. x + 2y = 0 (0,__), (4,__), (__,-3)

5. y = -3x (-2,__), (0,__), (__,-6)

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6. y = 3x – 6 (0,__), (__,0), (3,__)

7. 3x – 4y =12 (0,__), (__,0), (8,__)

8. -3x + y = 4 (1,__), (0,__), (-2,__)

9. 3x + 5y = 15 (0,__), (10,__),(__,0)

10. 2x – 5y = 10 (0,__), (10,__), (__,0)

B. Construct a table of ordered pairs. Use { -2, -1, 0, 1, 2} as values for x.

1. y = -x

x

y

2. y = 3x + 2

X

y

TOPIC 5.3.1 Graphs of Linear Equations in Two VariablesBased on Table of Ordered Pairs

OBJECTIVE

Draw the graph of Ax + By = C based on a table of values for x and y.

The graph of a linear equation Ax + By = C is the set of all points (x,y) that satisfies the linear equation in x and y.

Steps in constructing the graph of linear equations:

1. Assign values of x and compute the corresponding values of y.

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2. Set up a table values to obtain ordered pairs (x,y).3. Plot the points in the coordinate plane and connect these points with a straight line.

EXAMPLE:

Given a table of ordered pairs, construct the graph of y = 8 – 3x.

EXERCISES:

Draw the graph of the given tables below:

1. x 1 2 3 4 5y 1 2 3 4 5

x -2 -1 0 1 2

y 14 11 8 5 2

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2.

x -4 -2 0 2 4y 0 2 4 6 8

3.x -2 -1 0 1 2y 2 1 0 -1 -2

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4.

5.x -5 -4 -3 -2 -1y 3 7 6 5 4

TOPIC 5.3.2 Properties of Graphs of Linear Equations

OBJECTIVE

Determine the different properties of the graphs of a linear equation Ax + By = C.

The x-intercept of a line is the coordinate of the point where the line intersects the x-axis. The y-intercept of a line is the coordinate of the point where the line intersects the y-axis. If the graph of a linear equation trends to the right, then it is increasing and it has a positive slope. If the graph trends to the left, then it is decreasing and it has a negative slope.

EXAMPLES:

x -2 -1 0 1 2y -3 -1 1 3 5

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Given the graphs below, determine the x- and y- intercepts, trend (increasing or decreasing) and the slope ( positive or negative).

1. a. x –intercept: (-3,0)b. y-intercept: (0,2)

c. trend: increasingd. slope: positive

2. a. x-intercept: (-3,0)b. y-intercept: (0,-1)

c. trend: decreasingd.slope: negative

EXERCISES:

A. Determine the x-and y-intercepts, trend (increasing or decreasing) and the slope (positive or negative).

1.

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a. X-intercept: __________________b. Y-intercept:__________________c. Trend:_______________________d. Slope:_______________________

2.

a. X-intercept: __________________b. Y-intercept:__________________c. Trend:_______________________d. Slope:_______________________

3.

a. X-intercept: __________________

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b. Y-intercept:__________________c. Trend:_______________________d. Slope:_______________________

4.

a . X-intercept: __________________b. Y-intercept:__________________c. Trend:_______________________d. Slope:_______________________

B. Determine the x-and y-intercepts, trend (increasing or decreasing) and the slope (positive or negative).

1.

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a. X-intercept: __________________b. Y-intercept:__________________c. Trend:_______________________d. Slope:_______________________

2.

a. X-intercept: __________________b. Y-intercept:__________________c. Trend:_______________________d. Slope:_______________________

3.

a. X-intercept: __________________b. Y-intercept:__________________c. Trend:_______________________d. Slope:_______________________

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4.

a . X-intercept: __________________b. Y-intercept:__________________c. Trend:_______________________d. Slope:_______________________

TOPIC 5.3.3 Characteristics of the Graph of a Linear Equation

OBJECTIVE

Give the characteristics of the graph of a linear equation using the given linear equation.

Given the equation 2x + 5y = 10, determine the x-intercept, y-intercept, the slope, and the trend.

To find the x-intercept of the graph of an equation, we must let y = 0, then solve forthe value of x.

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2x + 5y = 102x + 5(0) = 102x + 0 = 10(1/2)2x = 10(1/2) X = 5 therefore, x-intercept is (5,0)

To find the y-intercept of the graph of an equation, let x = 0, then solve for the value of y.

2x + 5y =102(0) + 5y = 100 + 5y = 10(1/5)5y = 10(1/5) Y =2 therefore, y-intercept is (0,2)

To find the slope, transform the equation in the form y = mx + b where m is the slope.

2x + 5y = 10-2x + 2x + 5y = -2x + 10(1/5)5y = (-2x + 10)1/5 y = -2/5x + 2 therefore, since the slope is -2/5, the trend is decreasing.

EXERCISES:

A. Given the following equations, determine the x-intercept, y-intercept, slope and the trend.

x-intercept y-intercept slope trend1. X + 5y = 15 _________ __________ _________ ___________

2. 3x + y = 12 _________ __________ _________ ___________

3. 4x + 3y = 24 _________ __________ _________ ___________

4. 5x – 3y = 45 _________ __________ _________ ___________

5. -7x + 2y = 28 _________ __________ _________ ___________

6. 9x + 3y = -18 _________ __________ _________ ___________

7. -6x + 8y = 12 _________ __________ _________ ___________

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8. 10x – 3y = -20 _________ __________ _________ ___________

9. 15x + 12y =30 _________ __________ _________ ___________

10. 11x – 10y = 22 _________ __________ _________ ___________

B. Determine the x-intercept, y-intercept, slope and the trend of the following equations.

x-intercept y-intercept slope trend11. 5x + 9y=45 _________ __________ _________ ___________

12. 8x+ 3y=-48 _________ __________ _________ ___________

13. -12x+7y=28 _________ __________ _________ ___________

14. 8x+12y=-24 _________ __________ _________ ___________

15. 15x-10y=30 _________ __________ _________ ___________

16. -9x+4y=-72 _________ __________ _________ ___________

17. 4x+ 5y=5 _________ __________ _________ ___________

18. 3x-2y =12 _________ __________ _________ ___________

19. 2x + 3y= 4 _________ __________ _________ ___________

20. 5x-4y = 2 _________ __________ _________ ___________

TOPIC 5.3.4 Slope of a Line

OBJECTIVE

Find the slope of the line that passes through the given points.

A straight line may be determined by means of two distinct points. But a line can be

drawn also using one point of the line and its slope.

The slope of the line is the steepness of the line and is described as the ratio between

the change in y and the change in x.

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If P1( x1, y1) and P2 (x2,y2) are any two distinct points on a line, then the slope m of the line

is given by m = y1 – y2 (rise) , where x1 ≠ x2

X1 – x2 (run)

EXAMPLE 1:

Find the slope of the line that passes through (2,3) and (6,5).

Solution:

Let P1(x1,y1) = (2,3) and

P2(x2,y2) = (6,5)

Therefore;

m= y1-y2 = (3-5) = -2 or 1

x1-x2 (2-6) -4 2

m= 1

2

EXAMPLE2.

Find the slope of the line that passes through (3,5) and (5,2).

SOLUTION:

Let P1(x1,y1) = (3,5) and

P2(x2,y2) = (5,2)

Therefore;

m= y1-y2 = (5-2) = 3 or -3

x1-x2 (3-5) -2 2

m= -3

2

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Lines with a positive slope rise to the right. The larger the slope, the steeper the line rises.

Lines with a negative slope fall to the right.

Vertical lines which have equations of the form x = k, have undefined slope.

Horizontal lines which have equations of the form y = k, have a slope of zero.

EXERCISES:

A. Find the slope of the line that passes through the given points.

______________ 1. (2,6) and (3,9)

______________ 2. (-1,-3) and (7,3)

______________ 3. (4,8) and (5,8)

______________ 4. ( -6,1) and (-6,-6)

______________ 5. (0,0) and (1,3)

______________ 6. ( 0,0) and (-1, 3)

______________ 7. (-2,3) and (3,3)

______________ 8. ( -4,2) and (3,3)

______________ 9. (-2,1) and (6,4)

_______________ 10. (-4,-2) and (6,4)

B. Solve the following problems:

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1. Find the slope of the line that contains the points (1,2) and (4,5).

2. Find the slope of the line that passes through the points (-3,1) and (-2,-4).

3. Find the slope of the line with the following points (5,0) and (0,-4).

4. A line passes through points (4,-5) and (9,-3). Find the slope of the line.

5. Find the slope of the line through the points (4,6) and (0,4).

TOPIC 5.4 Rewriting Ax + By = C in the Formy = ax + b and Vice Versa

OBJECTIVE

Rewrite a linear equation Ax + By = C in the form y = ax + b and vice versa

A linear equation in two variables x and y is an equation of the form Ax + By = C where A, B, and C are real numbers and both A and B are not equal to zero. In this equation, the highest exponent of both variables x and y is 1. Hence, the linear equation is also known as the equation of the first degree.

A linear equation Ax + By = C can be written in the form y = ax + b.

EXAMPLE 1:

Write x – 3y = -6 in the form y = ax + b.

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SOLUTION:

x - 3y = -6 Givenx - x -3y = -6-x Subtract x from each side0 -3y = -6-x

(-1/3)-3y = -1/3(-6-x) Multiply -1/3 to each side Y = 2 + x Rearrange 2 + x

y = x+ 2

EXAMPLE 2:

Write y – 1 = x in the form Ax + By = C

SOLUTION:

y – 1 = x Given y -1 +1 = x +1 Addition Property of Equality y = x+ 1 y+(-x) = x+(-x) +1 Add –x to both sides y + (-x) = 1 Rearrange y + -x -x + y = 1

EXERCISES:

A. Transform Ax + By = C in the form of y = ax + b and vice versa.

1. 3x + 4y = 10

2. 2x – 5y = 6

3. 5x + 7y = -8

4. -4x + 3y = 9

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5. -2x + 3y = -7

6. y = 5x – 3

7. 2y = -4x + 5

8. -3y = 8x + 2

9. -2y = -7x + 9

10. -9y = 2x – 15

B. Rewrite Ax + By = C in the form y = ax + b and vice versa.

1. 10x + 7y = 9

2. 15x – 12y = 24

3. 1/2x + 5y = 3

4. 4x – 3/4y = 5

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5. 5/6x + 1/3y = 9

6. y = 13x – 9

7. y = -18x + 15

8. y = 5x/6 + 8

9. y = 3x + ¾

10. y = 2x + 10

5.4.1 Obtaining the Equation of the Line

OBJECTIVE

Obtain the equation of the line given the following from its graph;a. two points,b. slope and one point, andc. slope and y-intercept

1. Determine an equation of the line that passes through the points (1,3) and (-2,5).

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a. Get the slope using the formula m= y1-y2 = (3-5) = -2

x1-x2 (1- (-2)) 3

b. Determine the value of b by replacing the formula y = mx + b, by the obtained slope and one of the given points.

If point (1,3) If point (-2,5)y = mx + b y = mx + b3 = -2/3(1) + b 5= -2/3(-2) + b

2/3 +3 = -2/3 + 2/3 + b -(4/3) + 5= 4/3 +(-4/3) + b 3 2/3 or 11/3 = b 11/3= b

Note: You may use either of the points and will arrive at the same value of b as shown above.

c. Substitute the obtained value of slope and b in the equation y = mx + b and transform it in Ax + By = C

y = mx + by = -2/3x + 11/3

2/5x +y = -2/5x +2/5x +11/3 [2/5x +y = 11/3] 15

3(2x) + y = 5(11) Multiply the equation by its LCD to get the standard form. 6x + y = 55

2. Determine an equation of the line that passes through point (3,2) and has a slope of 3 . 4

a. Find the value of b by replacing the slope and the given point in the equation y = mx + b

y = mx + b2 = 3 (3) + b 42 = 9 + b 4(-9) + 2 = 9 + (-9) + b 4 4 4

-1 = b

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4

b. Substitute the slope and the obtained value of b in the equation y = mx + b and transform it in standard form.

y = mx + by = 3x – 1

4 4(-3x) + y = 3x + (-3x) – 1 4 4 4 4 -3x + y = -1 4 4[ -3x + y = -1 ] 4 4 4[ -3x + 4y = -1] -1 3x – 4y =1

c. Determine equation of the line with slope of -2 and y-intercept equal to 5 or (0,5).

Note that the y-intercept is also the value of b.

Substitute the slope with y-intercept in the equation y = mx + b and transform it in standard form.

y = mx + by = -2x + 5

2x + y = -2x +2x +5 2x + y = 5

EXERCISES:

A. Determine the equation of the line that satisfies the given requirements. Write the

equation in standard form Ax + By = C.

1. m = 3, y-intercept = -5

2. m =-2, y- intercept = 7

3. m= 1, y- intercept =8

4

4. m = -3, y –intercept = 10

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5

5. m = 5, and passes through the point (2,5)

6. m = -2, and passes through the point (-3,4)

7. m = 3, and passes through the point (-2,6)

4

8. m = 4, passes through the point (1,4)

5

9. Passes through the points (1,5) and (-3,9)

10. Passes through the points (-2,-3) and (-1,3)

B. Determine the equation of the line that satisfies the given requirements. Write the equation in standard form Ax + By = C.

1. Passes through the points (8,3) and (5,2)

2. Passes through the points (9,-2) and (4,3)

3. Passes through the points (7,5) and (-3,-1)

4. Passes through the points (-2,-3) and ( -1, 3)

5. m = 6, y-intercept = 23