Chapter 2 Solving Linear Equations. Mathematically Speaking 15x + 13y – 4(3x+2y) 15x + 13y – 12x...
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Transcript of Chapter 2 Solving Linear Equations. Mathematically Speaking 15x + 13y – 4(3x+2y) 15x + 13y – 12x...
![Page 1: Chapter 2 Solving Linear Equations. Mathematically Speaking 15x + 13y – 4(3x+2y) 15x + 13y – 12x - 8y 15x – 12x + 13y - 8y (15 – 12)x + (13 – 8)y 3x +](https://reader033.fdocuments.us/reader033/viewer/2022061420/56649ea05503460f94ba3408/html5/thumbnails/1.jpg)
Chapter 2
Solving Linear Equations
![Page 2: Chapter 2 Solving Linear Equations. Mathematically Speaking 15x + 13y – 4(3x+2y) 15x + 13y – 12x - 8y 15x – 12x + 13y - 8y (15 – 12)x + (13 – 8)y 3x +](https://reader033.fdocuments.us/reader033/viewer/2022061420/56649ea05503460f94ba3408/html5/thumbnails/2.jpg)
Mathematically Speaking
15x + 13y – 4(3x+2y)
15x + 13y – 12x - 8y
15x – 12x + 13y - 8y
(15 – 12)x + (13 – 8)y
3x + 5y
Can you identify what happens in each step?
![Page 3: Chapter 2 Solving Linear Equations. Mathematically Speaking 15x + 13y – 4(3x+2y) 15x + 13y – 12x - 8y 15x – 12x + 13y - 8y (15 – 12)x + (13 – 8)y 3x +](https://reader033.fdocuments.us/reader033/viewer/2022061420/56649ea05503460f94ba3408/html5/thumbnails/3.jpg)
Can you identify what has happened in each step?
15x + 13y – 4(3x+2y)
15x + 13y – 12x - 8y
15x – 12x + 13y - 8y
(15 – 12)x + (13 – 8)y
3x + 5y
- Given
-Distributive
-Commutative
-Factor
-Addition
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Identify the steps used to solve the equation, m + 4 = 29.
m+4=29
- 4=-4
m =25
Given
Inverse + -
Evaluate
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Identify the steps used to solve the equation.
3x + 4 = 19
- 4 = - 4
3x = 15
3= 3
x = 5
GivenInverse + EvaluateInverse *
Evaluate
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Identify the steps used to solve the equation.
5x – 4 = 2(x – 4) + 185x – 4 = 2x – 8 + 185x – 4 = 2x + 10 3x = 14
3
14x
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Identify the steps used to solve the equation.
5x – 4 = 2(x – 4) + 185x – 4 = 2x – 8 + 185x – 4 = 2x + 10 3x = 14
3
14x
• Given
Distributive
AdditionInverse Ops Like terms Inverse Ops
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Identify the steps used to solve the equation.
-5x + 3 + 2x = 7x – 8 + 9x
-3x +3 = 16x -8
11 = 19x
x19
11
19
11x
![Page 9: Chapter 2 Solving Linear Equations. Mathematically Speaking 15x + 13y – 4(3x+2y) 15x + 13y – 12x - 8y 15x – 12x + 13y - 8y (15 – 12)x + (13 – 8)y 3x +](https://reader033.fdocuments.us/reader033/viewer/2022061420/56649ea05503460f94ba3408/html5/thumbnails/9.jpg)
Identify the steps used to solve the equation.
-5x + 3 + 2x = 7x – 8 + 9x
-3x +3 = 16x -8
11 = 19x
x19
11
19
11x
Like Terms
Inverse Ops
Inverse Ops
Symmetric
property
Given
![Page 10: Chapter 2 Solving Linear Equations. Mathematically Speaking 15x + 13y – 4(3x+2y) 15x + 13y – 12x - 8y 15x – 12x + 13y - 8y (15 – 12)x + (13 – 8)y 3x +](https://reader033.fdocuments.us/reader033/viewer/2022061420/56649ea05503460f94ba3408/html5/thumbnails/10.jpg)
So what is the definition? Which of these equations are linear?
x+y = 5
2x+ 3y = 4
7x-3y = 14
y = 2x-2
y=4
x2 + y = 5
x = 5
xy = 5
x2 +y2 = 9
y = x2
3
y
Linear Not Linear
The degree must be one.
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2.1 What is a solution?
What happens when one solves an equation?
You might say “One gets an answer.”
What is the format of that answer?
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What happens when one solves an equation?
1. The solutions is a Unique solution.
2. The solution is Infinite solutions.
3. The is no possible solution.
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What happens when one solves an equation?
1. The solution is a Unique solution.• There is only ONE numerical answer to
solve the equation.
2. The solution is Infinite solutions. • IDENTITY. The equations are
mathematically equivalent.
3. There is no possible solution.• INCONSISTENT. With linear equations
this means there is no point of intersection.
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2.2 One linear equation in one variable
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One Solution.
3x + 4 = 19
- 4 = - 4
3x = 15
3= 3
x = 5
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Infinite Solutions. IDENTITY
14 + 5x – 4 = (x + 4x)-8 + 18
14 + 5x – 4 = 5x – 8 + 18 5x + 10 = 5x + 10 10 = 10
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No Solution. INCONSISTENT
-7x + 3 + 1x = 2x – 8 - 8x
-6x +3 = -6x -8
3 = -8
83
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2.3 Several linear equations in one variable
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Systems of Equations
Solving systems of equations with two or more linear equations
Substitution
Elimination
Cramer’s Rule
Graphical Representation
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The 3 possible solutions still occur.
1. The solution is a Unique solution.• This one solution is in the form of a
point. (e.g. (x,y), (x,y,z) )
2. The solution is Infinite solutions. • IDENTITY. The lines are the same line.
3. There is no possible solution.• INCONSISTENT. The lines are
parallel (2-D) or skew (3-D).
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Substitution – use substitution when…
One of the equations is already solved for a variable.
y = 2x – 53x + 4y = 13
Substitute the first equation into the second3x + 4(2x – 5) = 13
Solve for the variable3x + 8x – 20 = 1311x = 33x = 3
Substitute back into one of the original equations y = 2(3) – 5 = 1 Final Answer (3,1)
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Elimination – use elimination when substitution is not set up.
Elimination ELIMINATES a variable through manipulating the equations.
Some equations are setup to eliminate.
Some systems only one equation must be manipulated
Some systems both equations must be manipulated
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Setup to Eliminate
Given2x – 4y = 8
3x + 4y = 2The y terms are opposites, they will eliminateAdd the two equations 5x = 10 x = 2Substitute into an original equation
3(2) + 4y = 2 6 + 4y = 2 4y = -4 y = -1
Final Answer (2,-1)
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Manipulate ONE eqn. to Eliminate
Given2x + 2y = 8
3x + 4y = 2Multiply the first equation by – 2 to elim. y terms-4x – 4y = -16
3x + 4y = 2Add the two equations -1x = -14 x = 14Substitute into an original equation 3(14) + 4y = 2 42 + 4y = 2 4y = -40 y = -10
Final Answer (14,-10)
![Page 25: Chapter 2 Solving Linear Equations. Mathematically Speaking 15x + 13y – 4(3x+2y) 15x + 13y – 12x - 8y 15x – 12x + 13y - 8y (15 – 12)x + (13 – 8)y 3x +](https://reader033.fdocuments.us/reader033/viewer/2022061420/56649ea05503460f94ba3408/html5/thumbnails/25.jpg)
Manipulate BOTH eqns. to Eliminate
Given2x + 3y = 4
3x + 4y = 2Multiply the first equation by 3 & the second equation by -2 to elim. x terms
6x + 9y = 12 -6x - 8y = -4
Add the two equations y = 8Substitute into an original equation
2x + 3(8) = 4 2x + 24 = 4 2x = -20 x = -10
Final Answer (-10,8)
![Page 26: Chapter 2 Solving Linear Equations. Mathematically Speaking 15x + 13y – 4(3x+2y) 15x + 13y – 12x - 8y 15x – 12x + 13y - 8y (15 – 12)x + (13 – 8)y 3x +](https://reader033.fdocuments.us/reader033/viewer/2022061420/56649ea05503460f94ba3408/html5/thumbnails/26.jpg)
Identity Example
2x + 3y = 12
y = -2/3 x + 4
Using substitution
2x + 3(-2/3 x + 4) = 12
2x – 2x + 12 = 12
12 = 12
Identity
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Inconsistent Example
3x – 4y = 18
3x – 4y = 9
Use Elimination by multiplying Eqn 2 by -1.
3x – 4y = 18
-3x + 4y = -9
0 = 9 False
Inconsistent
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3 Equations: 3 Variables required
Eqn1: 3y – 2z = 6Eqn2: 2x + z = 5Eqn3: x + 2y = 8
Solve Eqn2 for zz = -2x + 5
Now substitute into Eqn13y – 2(-2x+5) = 63y + 4x – 10 = 6
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3 Equation continued…
NEW: 4x + 3y = 16
Eqn3: x + 2y = 8
Now one can either substitute or eliminate
NEW: 4x + 3y = 16
Eqn3(*-4): -4x - 8y = -32
-5y = -16
y = 16/5
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Now having a value for y, one can substitute into x + 2(-16/5) = 8x = 8 + 32/5 = 40/5 + 32/5
x = 72/5
This can now be substituted into our Eqn2 solved for z
z = - 2(72/5) + 5
z = -144/5 + 5 = -144/5 + 25/5
z = -119/5 Final Answer(72/5, -16/5, -119/5)
And still continued…
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Matrices: Cramer’s Rule
Dimensions: row x columns
Determinant
a bc d
ad - bc
ef
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Cramer’s Rule set up
e bf d
a eb fx = y =
determinant determinant
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Example
2x + 3y = 5
4x + 5y = 7
The determinant is 10-12 = -2
2 34 5
57
5 37 5
2 54 7x = y =
-2 -2
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Solve for x and y…
4/-2
x = -2
-6/-2
y = 3
5 37 5
2 54 7
x setup y setup
-2 -2
2
2125
2
2014
Final answer (-2,3)
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You cannot use Cramer’s Rule if the difference of the products is 0.
![Page 36: Chapter 2 Solving Linear Equations. Mathematically Speaking 15x + 13y – 4(3x+2y) 15x + 13y – 12x - 8y 15x – 12x + 13y - 8y (15 – 12)x + (13 – 8)y 3x +](https://reader033.fdocuments.us/reader033/viewer/2022061420/56649ea05503460f94ba3408/html5/thumbnails/36.jpg)
Verbal Models
Verbal Models are math problems written in word form
General Rule: Like reading English -Left to Right
Special Cases: Change in order terms some time called “turnaround” words (Cliff Notes: Math Word Problems, 2004)
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Convert into Math…
Two plus some number
A number decreased by three
Nine into thirty-six
Seven cubed
Eight times a number
Ten more than five is what number
2+x
x-3
36 / 9
7^3 73
8x
5 + 10 = x
into
more than
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MORE Convert into Math…
Twenty-five percent of what number is twenty-two?
The quantity of three times a number divided by seven equals nine.
The sum of two consecutive integer is 23.
.25 * x = 22
(3x)/7 = 9
x + (x+1) = 23
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Work Problem.
I can mow the yard in 5 hours. My husband can mow the yard in 2 hours. If we mowed together how long would it take for us to mow the yard.
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SolutionMy rate is 1 yard per 5 hours: 1/5 t
Doug’s rate is 1 yard per 2 hours; ½ t
together = addition
The whole job = 1
the common denominator is 10
Solve for t
7t = 10; t = 10/7 or 1.42857 ish
12
1
5
1 tt
1052 tt
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Formulas you should know…
Area of Rectangle
Perimeter of Rectangle
Area of Triangle
Area of Circle
A = hb
P = 2 (h + b)
A = ½ hb
A = r2
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Candy
I bought 3 bags of candy and 5 chocolate bars. I spent $13. My friend spent $17 and she bought 4 bags of candy and 6 chocolate bars. What is the cost of the candy bags and chocolate bars?
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Solution3b + 5c = 134b + 6c = 17
det = 18-20 = -2x = y =
x = (78-85)/-2 y = (51-52)/-2x = -7/2 = $3.50 y = -1/-2 = 0.50
3 54 6
1317
13 517 6
-2
3 134 17
-2