LT 8 Systems of Equations
May 7th, 2013 - May 10th, 2013
May 7th, 2013
Objective: Solve Linear systems using elimination (to find the coordinates that both equations share)
Skill #1 Using Addition to eliminate a variable.
Example #1 2x + 3y = 11
-2x + 5y = 13
Step #1 Add the equations to eliminate one variable.
Step #2 Solve the new equation for the remaining variable.
Step #3 Substitute the solution in one of the original equations and solve for the other variable.
Check the solution by plugging in the coordinates (X, Y) into both equations:
2x + 3y = 11 -2x + 5y = 13
1

Example #2 4x - 3y = 5
-2x + 3y = -7
Step #1 Add the equations to eliminate one variable.
Step #2 Solve the new equation for the remaining variable.
Step #3 Substitute the solution in one of the original equations and solve for the other variable.
Check the solution by plugging in the coordinates (X, Y) into both equations:
4x - 3y = 5 -2x + 3y = -7
2

Skill #2 Use Subtraction to eliminate a variable.
Example #1 4x + 3y = 2
5x + 3y = -2
Step #1 Subtract the equations to eliminate one variable.
Step #2 Solve the new equation for the remaining variable.
Step #3 Substitute the solution in one of the original equations and solve for the other variable.
Check the solution by plugging in the coordinates (X, Y) into both equations:
4x + 3y = 2 5x + 3y = -2
3

Example #1 7x - 2y = 5
7x - 3y = 4
Step #1 Subtract the equations to eliminate one variable.
Step #2 Solve the new equation for the remaining variable.
Step #3 Substitute the solution in one of the original equations and solve for the other variable.
Check the solution by plugging in the coordinates (X, Y) into both equations:
7x - 2y = 5 7x - 3y = 4
4

Skill #3 What if the equation is in the wrong order to add or subtract?
Example #1 8x -4y = -4
4y = 3x = 14
Step #1 Write Equation 2 so that the like terms are arranged in columns.
Step #2 Add the equations to eliminate one variable.
Step #3 Solve the new equation for the remaining variable.
Step #4 Substitute the solution in one of the original equations and solve for the other variable.
5

Rewrite the linear system so that the like terms are arranged in columns.
1.
8x –y = 19
y + 3x = 7
2.
4x = y –11
6y + 4x = –3
3.
9x –2y = 5
2y = –11x + 8
Solve the linear system by using elimination.
4.
x + 5y = –8
–x –2y = –13
5.
7x – 4y = –30
3x + 4y = 10
6

7
6.
6x + y = 39
–2x + y = –17
7.
3x = y –20
–7x 2y = 40
8.
2x –6y = –10
4x = 10 + 6y
9.
x –3y = 6
–2x = 3y + 33
10.
–3x = y –20
–y = –5x + 4

12.
Fishing Barge A fishing barge leaves from a dock and moves upstream (against the current) at a rate of 3.8 miles per hour until it reaches its destination. After the people on the barge are done fishing, the barge moves the same distance downstream (with the current) at a rate of 8 miles per hour until it returns to the dock. The speed of the current remains constant. Use the models below to write and solve a system of equations to find the average speed of the barge in still water and the speed of the current.
Upstream: Speed of barge in still water – Speed of current = Speed of barge
Downstream: Speed of barge in still water + Speed of current = Speed of barge
13.
Floor Sander Rental A rental company charges a flat fee of x dollars for a floor sander rental plus y dollars per hour of the rental. One customer rents a floor sander for 4 hours and pays $63.
Another customer rents a floor sander for 6 hours and pays $87. a Find the flat fee and the cost per hour for the rental. b.
How much would it cost someone to rent a sander for 11 hours?
8

May 9th, 2013
Objective: Solve Linear systems by Multiplying First to find the coordinates both equations share.
Skill #1 Multiple one equation, then add/subtract.
Example 1: 6x + 5y = 19
2x + 3y = 5
Step 1: Multiply second equation so that one of the variables cancels out when added/subtracted together.
Step 2: Add/subtract the equations:
Step 3: Solve the new equation for the remaining variable.
Step #4 Substitute the solution in one of the original equations and solve for the other variable.
Step 5: Plug the coordinates back into the original equations:
6x + 5y = 19 2x + 3y = 5
9

Example 2: 6x - 2y = 1
-2x + 3y = -5
Step 1: Multiply second equation so that one of the variables cancels out when added/subtracted together.
Step 2: Add/Subtract the equations:
Step 3: Solve the new equation for the remaining variable.
Step #4 Substitute the solution in one of the original equations and solve for the other variable.
Step 5: Plug the coordinates back into the original equations:
6x - 2y = 1 -2x + 3y = -5
10

Skill #2 Multiple both equations, then subtract/add.
Example 1: 4x + 5y = 35
2y = 3x - 9
Step 1: Rearrange the second equation so it is in columns with the first.
Step #2 : Multiple both equations by a coefficient so that the x's or y's will cancel each other out by adding/subtracting.
Step 3: Add/Subtract the equations:
Step 4: Solve the new equation for the remaining variable.
Step 5 Substitute the solution in one of the original equations and solve for the other variable.
Step 6: Plug the coordinates back into the original equations:
4x + 5y = 35 2y = 3x - 9
11

Example 2: 7x + 3y = -12
2x + 5y = 38
Step #1: Multiple both equations by a coefficient so that the x's or y's will cancel each other out by adding/subtracting.
Step 2: Add/Subtract the equations:
Step 3: Solve the new equation for the remaining variable.
Step #4 Substitute the solution in one of the original equations and solve for the other variable.
Step 5: Plug the coordinates back into the original equations:
7x + 3y = -12 2x + 5y = 38
12

Practice May 10, 2013
Describe the first step you would use to solve the linear system.
1.
3x – 4y = 7
5x + 8y = 10
2.
9x + 4y = 13
3x + 5y = 9
3.
5x + 7y = –3
15x + 4y = –5
Solve the linear system by using elimination.
4.
x + 3y = 1
–5x + 4y = –24
5.
–3x – y = –15
8x + 4y = 48
6.
x + 7y = –37
2x – 5y = 21
13

7.
8x – 4y = –76
5x + 2y = –16
8.
–3x + 10y = 23
5x + 2y = 55
9.
9x – 4y = 26
18x + 7y = 22
10.
4x – 3y = 16
16x + 10y = 240
11.
20x 10y = 100
–5x + 4y = 53
14

12.
3x – 10y = –25
5x – 20y = –55
13.
–3x – 4y = 27
5x – 6y = –7
14.
Travel Agency A travel agency offers two Chicago outings. Plan A includes hotel accommodations for three nights and two pairs of baseball tickets worth a total of $557. Plan B includes hotel accommodations for five nights and four pairs of baseball tickets worth a total of $974.
Let x represent the cost in dollars of one night’s hotel accommodations and let y represent the cost in dollars of one pair of baseball tickets. a.
Write a linear system you could use to find the cost of one night’s hotel accommodations and the cost of one pair of baseball tickets. b.
Solve the linear system to find the cost of one night’s hotel accommodations and the cost of one pair of baseball tickets.
15.
Highway Project There are fifteen workers employed on a highway project, some at $180 per day and some at $155 per day. The daily payroll is $2400. Let x represent the number of $180 per day workers and let y represent the number of $155 per day workers. Write and solve a linear system to find the number of workers employed at each wage.
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