2-1 Solving One-Step Equations Warm Up Warm Up Lesson Presentation Lesson Presentation California...

2-1 Solving One-Step Equations

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California Standards

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Warm UpEvaluate.

1. – + 4

3. 0.96 ÷ 6

Evaluate each expression for a = 3 and b = –2.

6. a + 5 7. 12b

–0.813

23

3 23

–24 8

2. –0.51 + (–0.29)

4. (–9)(–9)0.16 81

5. 1


Preparation for 5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable, and provide justification for each step.Also covered: 2.0

California Standards


equationsolution of an equationsolution set

Vocabulary


An equation is a mathematical statement that two expressions are equal.

A solution of an equation is a value of the variable that makes the equation true. A solution set is the set of all solutions. Finding the solutions of an equation is also called solving the equation.


Inverse Operations

Add x. Subtract x.

Multiply by x. Divide by x.

An equation is like a balanced scale. To keep the balance, you must perform the same inverse operation on both sides.

To find solutions, perform inverse operations until you have isolated the variable. A variable is isolated when it appears by itself on one side of an equation, and not at all on the other side.


Solution sets are written in set notation using braces, { }. Solutions may be given in set notation, or they may be given in the form x = 14.

Writing Math


Solve the equation.

Additional Example 1A: Solving Equations by Using Addition or Subtraction

Since 8 is subtracted from y, add 8 to both sides to undo the subtraction.

y – 8 = 24 + 8 + 8

y = 32

Check y – 8 = 24

32 – 8 2424 24

To check your solution, substitute 32 for y in the original equation.

The solution set is {32}.


Solve the equation.

Additional Example 1B: Solving Equations by Using Addition

To check your solution, substitute 2.4 for t in the original equation.

Since 1.8 is added to t, subtract 1.8 from both sides to undo the addition.

4.2 = t + 1.8 –1.8 –1.8

2.4 = t

Check 4.2 = t + 1.8

4.2 2.4 + 1.8

4.2 4.2

The solution set is {2.4}.


Solve the equation. Check your answer.

Check It Out! Example 1a

Since 3.2 is subtracted from n, add 3.2 to both sides to undo the subtraction.

n – 3.2 = 5.6

+ 3.2 + 3.2

n = 8.8

Check n – 3.2 = 5.6

8.8 – 3.2 5.65.6 5.6

To check your solution, substitute 8.8 for n in the original equation.




Check It Out! Example 1b

Since 6 is subtracted from k, add 6 to both sides to undo the subtraction.

–6 = k – 6

+ 6 + 6

0 = k

Check –6 = k – 6

–6 0 – 6–6 –6

To check your solution, substitute 0 for k in the original equation.




Check It Out! Example 1c

Since 6 is added to t, subtract 6 from both sides to undo the addition.

6 + t = 14

– 6 – 6

t = 8

Check 6 + t = 14

6 + 8 1414 14

To check your solution, substitute 8 for t in the original equation.




Additional Example 2A: Solving Equations by Using Multiplication or Division

Since j is divided by 3, multiply from both sides by 3 to undo the division.

–8 –8

To check your solution, substitute –24 for j in the original equation.

–24 = j

Check

The solution set is {–24}.



Additional Example 2B: Solving Equations by Using Multiplication or Division

Since v is multiplied by –6, divide both sides by –6 to undo the multiplication.

–4.8 = –6v

0.8 = v

Check –4.8 = –6v –4.8 –6(0.8)

–4.8 –4.8

To check your solution, substitute 0.8 for v in the original equation.



Solve each equation. Check your answer.


Since p is divided by 5, multiply both sides by 5 to undo the division.

To check your solution, substitute 50 for p in the original equation.

p = 50

Check

10 10




Check It Out! Example 2b

Since y is multiplied by 0.5, divide both sides by 0.5 to undo the multiplication.

0.5y = –10

Check 0.5y = –10

0.5(–20) –10

–10 –10

To check your solution, substitute –20 for y in the original equation.

y = –20 The solution set is {–20}.




Since c is divided by 8, multiply both sides by 8 to undo the division.

c = 56

To check your solution, substitute 56 for c in the original equation.

7 7


Check


When solving equations, you will sometimes find it easier to add an opposite to both sides instead of subtracting or to multiply by a reciprocal instead of dividing. This is often true when an equation contains negative numbers or fractions.


Additional Example 3A: Solving Equations by Using Opposites or Reciprocals

Solve each equation.

The reciprocal of is . Since

w is multiplied by multiply

both sides by .

The solution set is {–24}.


Additional Example 3B: Solving Equations by Using Opposites or Reciprocals


Since p is added to , add to both sides to undo the subtraction.

The solution set is .{ }




Since –2.3 is added to m, add 2.3 to both sides.

–2.3 + m = 7+2.3 + 2.3

m = 9.3

–2.3 + m = 7


To check your solution, substitute 9.3 for m in the original equation.

Check –2.3 + m = 7–2.3 + 9.3 7

7 7


5 4

+ z = Since is added to z add

to both sides.


Solve the equation. Check your answer. Check It Out! Example 3b

To check your solution, substitute 2 for z in the original equation.

Check



w = 612


The reciprocal of is . Since

w is multiplied by multiply

both sides by .

Check

102 102

To check your solution, substitute 612 for w in the original equation.



Additional Example 4: Application

Ciro deposits of the money he earns from mowing lawns into a college education fund. This year Ciro added $285 to his college education fund. Write and solve an equation to find out how much money Ciro earned mowing lawns this year.

14


Additional Example 4 Continued

e = $1140

The original earnings were $1140 .

Write an equation to represent the relationship.

earnings is

times $285 14

14

41 e = 285

41 The reciprocal of is . Since e

is multiplied by ,

multiply both sides by

1414

41

41

.

e = $285


Check It Out! Example 4

The distance in miles from the airport that a plane should begin descending divided by 3 equals the plane’s height above the ground in thousands of feet. A plane is 10,000 feet above the ground. Write and solve an equation to find the distance from the airport at which this plane should begin descending.


Check It Out! Example 4 Continued

d ÷ 3 = h

Write an equation to represent the relationship.

d = 30

At 10,000 feet altitude the decent should start 30,000 feet from the airport.

distancedivided by

heightis3

31 d

3= 103

1 Substitute 10 for h. The reciprocal of

is . Since d is multiplied by

multiply both sides by .

13

31

13

31


Lesson Quiz


1. r – 4 = –8

3.

5.

6. This year a high school had 578 sophomores enrolled. This is 89 less than the number enrolled last year. Write and solve an equation to find the number of sophomores enrolled last year.

–4

2.8

s – 89 = 578; s = 667

2.

4. 8y = 4

40

2-1 Solving One-Step Equations Warm Up Warm Up Lesson Presentation Lesson Presentation California...

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