systems of linear equations 11/11/12 lntaylor ©. table of contents learning objectives solving...

Systems of Linear Equations

11/11/12 lntaylor ©

Table of Contents

Learning Objectives

Solving Systems by Graphing

Solving Systems by Substitution

Solving Systems by Elimination

Word Problems

Practice

11/11/12 lntaylor ©

Learning Objectives

TOC11/11/12 lntaylor ©

Solving Systems of Linear Equations

• In these sections you will learn/review how to:

– Meet or exceed proficiency in AF 9.0

– Pass questions regarding these standards on District or State Tests

– Solve Systems of Equations using three (3) different methods


1st MethodGraphing



• Graphing Method

– Graph two different linear equations– Determine the point where the lines crosses– Prove that point works with both equations


Find the solution to:

y = 2x + 1 and

y = - x + 4


0,0

Find the solution to y = 2x + 1 and y = - x + 4

Step 1 – Graph the 1st lineIdentify slope m and bLocate 0,0Go up or down b Put a dot for the 1st pointGo up or down ΔyGo right ΔxPut a dot for 2nd pointConnect dotsLabel the line

y = 2x + 1


Step 2 – Graph the 2nd lineIdentify slope m and bLocate 0,0Go up or down b Put a dot for the 1st pointGo up or down ΔyGo right ΔxPut a dot for 2nd pointConnect dotsLabel the lineFind the point where the lines crossPlug into each equationThe point (1,3) is the solution for both equations

y = - x + 4

y = 2x + 1 and y = -x + 43 = 2(1) + 1 and 3 = -1 + 43 = 3 and 3 = 3

(1,3)

Now you try!

y = 1/2x + 1 and

y = - x + 4


0,0

Find the solution to y = 1/2x + 1 and y = - x + 4


y = 1/2x + 1



y = - x + 4

y = 1/2x + 1 and y = - x + 42 = 1/2(2) + 1 and 2 = - 2 + 42 = 2 and 2 = 2

(2,2)

Now you try!

y = 1/3x – 4and

y = - x


0,0

Find the solution to y = 1/3x - 4 and y = - x


y = 1/3x - 4



y = - x + 4

y = 1/3x – 4 and y = - x-3 = 1/3(3) – 4 and - 3 = - 3- 3 = - 3 and - 3 = - 3

(3, - 3)

2nd MethodSubstitution



• Substitution Method

– Set one equal to y – Substitute the 2nd equation into the 1st equation



y = 2x + 1 and

y = - x + 4


Step 4

Step 5

Rewrite without the y and solve

Plug the answer into both equations

Step 6 The 1st number is x; the 2nd number is y

Step 7 The solution is (x,y) or in this equation (1,3)

Step 2

Step 1 Write the 1st equation as y =

Write the 2nd equation as = y

Step 3 They are both = y

11/11/12 lntaylor ©TOC

Find the solution to y = 2x + 1 and y = - x + 4y = 2x + 1 - x + 4 = y

- x + 4 = 2x + 1 4 – 1 = 2x + x 3 = 3x x = 1

- x + 4 = 2x + 1- (1) + 4 = 2(1) + 1 - 1 + 4 = 2 + 1 3 = 3

(1,3)

Now you try!

y = 1/2x + 1 and

y = - x + 4


Step 4

Step 5





Step 2





Find the solution to y = 1/2x + 1 and y = - x + 4y = 1/2x + 1 - x + 4 = y

- x + 4 = 1/2x + 1 4 – 1 = 1/2x + x 3 = 1½ x 6 = 3x x = 2

- x + 4 = 1/2x + 1- (2) + 4 = ½(2) + 1 - 2 + 4 = 1 + 1 2 = 2(2,2)

Now you try!

y = 1/3x – 4and

y = - x


Step 4

Step 5




Step 7 The solution is (x,y) or in this equation (3,- 3)

Step 2





Find the solution to y = 1/3x - 4 and y = - x y = 1/3x - 4 - x = y

- x = 1/3x - 4 4 = 1/3x + x 4 = 1⅓ x 12 = 4x x = 3

- x = 1/3x - 4- (3) = ⅓(3) - 4 - 3 = 1 - 4 - 3 = - 3(3,- 3)

3rd MethodElimination



• Elimination Method

– Write the equations underneath each other – Choose a multiplier for one equation that eliminates a

variable– Plug in the answer and solve for the 2nd variable



y = 2x + 1 and

y = - x + 4


Step 4

Step 5

Add the equations and solve for the variable

Plug the answer into an equation and solve

Step 6 Make sure you put the numbers in the right order!


Step 2

Step 1 Write the equations underneath each other

Pick a multiplier that will eliminate 1 variable

Step 3 Rewrite the equations


Find the solution to y = 2x + 1 and y = - x + 4y = 2x + 1 y = - x + 4

(2) y = (2)(-x + 4) 2y = - 2x + 82y = - 2x + 8

3y = 9 y = 3 (x,3)

y = 2x + 13 = 2x + 12 = 2xx = 1

(1,3)

Now you try!

y = 1/2x + 1 and

y = - x + 4


Step 4

Step 5





Step 2





Find the solution to y = ½x + 1 and y = - x + 4y = ½x + 1 y = - x + 4

(2) y = (2)(½x + 1) 2y = x + 22y = x + 2

3y = 6 y = 2 (x,2)

y = ½x + 12 = ½x + 11 = ½x2 = x

(2,2)

Now you try!

y = 1/3x – 4and

y = - x


Step 4

Step 5




Step 7 The solution is (x,y) or in this equation (3,- 3)

Step 2





Find the solution to y = ⅓x - 4 and y = - xy = ⅓x - 4 y = - x

(3) y = (3)(⅓x - 4) 3y = x - 123y = x - 12

4y = - 12 y = - 3 (x,- 3)

y = ⅓x - 4- 3 = ⅓x - 4 1 = ⅓x 3 = x

(3,- 3)

Word Problems


Step 4

Step 5


Add the equations and solve

Step 6 Plug in the answer and solve

Step 2

Step 1 Highlight the first equation

Highlight the second equation



4A + 9C = $824A + 1C = $34 – 1(4A + 1C) = – 1($34)

– 4A – 1C = - $34

4A + 9C = $82– 4A – 1C = – $34

8C = $48 C = $6

4A + 9C = $824A + 9($6) = $824A + $54 = $82 4A = $28 A = $7

Adult Tickets were $7Child Tickets were $6

Step 4

Step 5

Pick multiplier(s) that will eliminate 1 variable


Step 6 Plug into the original equation and solve

Step 2





5A + 7C = $1296A + 10C = $174

6(5A + 7C) = 6($129) or 30A + 42C = $774 30A + 42C = $774– 30A – 50C = – $870

– 8C = – $96 C = $12

5A + 7C = $1295A + 7($12) = $1295A + $84 = $129 5A = $45 A = $9

Adult Tickets were $9Child Tickets were $12

– 5(6A + 10C) = – 5($174) or – 30A – 50C = – $870

Step 4

Step 5

Pick multiplier(s) that will eliminate 1 variable



Step 2





3V + 6B = 19210V + 5B = 205

10(3V + 6B) = 10(192) or 30V + 60B = 1920 30V + 60B = 1920– 30V – 15B = – 615

45B = 1305 B = 29

3V + 6B = 192 3V + 6(29) = 192 3V + 174 = 192 3V = 18 V = 6

6 students in each van29 students in each bus

– 3(10V + 5B) = – 3(205) or – 30V – 15B = – 615

Step 4

Step 5

No multiplier(s) is needed here!



Step 2





A + B = 16 A – B = 2

2A = 18 A = 9

A + B = 16 9 + B = 16 B = 7 A is 9

B is 7

Practice



Problem Answer

y = 2x + 5 and y = – 5x – 16

y = – 4x + 5 and y = – 2x + 3

y = 5x + 10 and 7x – 7y = - 14

y = – 6x + 20 and – x – y = - 10

y = 5x – 5y = – 20 and x + 2y = 17

x – 4y = 16 and – 3x – 3y= 12

– 5x – 4y = – 22 and – 3x – y = – 16

– x – 5y = – 3 and – 8x – 6y = 10

2x – 7y = 6 and – 7x + 5y = – 21

> (-3, -1) >

>

>

>

>

(1, 1)(- 2, 0)

(2, 8)

(3, 7)

(0, - 4)

> (6, - 2)

> (- 2, 1)

> (3, 0)

clear answers

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