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 + 5y

Can you identify what happens in each step?

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

Identify the steps used to solve the equation, m + 4 = 29.

m+4=29

- 4=-4

m =25

Given

Inverse + -

Evaluate

Identify the steps used to solve the equation.

3x + 4 = 19

- 4 = - 4

3x = 15

3= 3

x = 5

GivenInverse + EvaluateInverse *

Evaluate

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

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

Identify the steps used to solve the equation.

-5x + 3 + 2x = 7x – 8 + 9x

-3x +3 = 16x -8

11 = 19x

x19

11

19

11x

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

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.

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?

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.

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.

2.2 One linear equation in one variable

One Solution.

3x + 4 = 19

- 4 = - 4

3x = 15

3= 3

x = 5

Infinite Solutions. IDENTITY

14 + 5x – 4 = (x + 4x)-8 + 18

14 + 5x – 4 = 5x – 8 + 18 5x + 10 = 5x + 10 10 = 10

No Solution. INCONSISTENT

-7x + 3 + 1x = 2x – 8 - 8x

-6x +3 = -6x -8

3 = -8

83

2.3 Several linear equations in one variable

Systems of Equations

Solving systems of equations with two or more linear equations

Substitution

Elimination

Cramer’s Rule

Graphical Representation

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).

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)

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

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)

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)

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)

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

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

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

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

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…

Matrices: Cramer’s Rule

Dimensions: row x columns

Determinant

a bc d

ad - bc

ef

Cramer’s Rule set up

e bf d

a eb fx = y =

determinant determinant

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

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)

You cannot use Cramer’s Rule if the difference of the products is 0.

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)

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

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

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.

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

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

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?

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