# Solving Linear Systems by Linear Combinations Worksheet

Solving Linear Systems by Linear Combinations Worksheet
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1.
Sheela wanted to borrow 100 wine glasses from Emily for a party. Emily decided to send them through her servant, Tim. She offered him incentive of 30 cents for every glass delivered safely and threatened to forfeit 90 cents for every glass he broke. On settlement Tim received \$24 from Emily. How many glasses did Tim break?
 a. 15 glasses b. 10 glasses c. 25 glasses d. 5 glasses

#### Solution:

Let x be the number of glasses delivered intact then 30x is the amount earned.
[Analyse from the data given.]

Let y be the number of glasses broken then 90y is the amount forfeited.
[Analyse from the data provided.]

x + y = 100 ---------- (1)
[According to the data.]

30x - 90y = 2400
30(x - 3y) = 2400
x - 3y = 2400 / 30
[According to the data.]

x - 3y = 80 ---------- (2)
[According to the data.]

3x + 3y = 300
x   - 3y = 80
____________
4x    =  380
[Multiply equation (1) with 3 and add it to equation (2).]

x = 95
[Simplify.]

y = 100 - 95 = 5
[Substitute value of x in equation (1) to find y.]

Therefore, Tim has broken 5 glasses.

2.
Which of the following solutions satisfies the linear system?
2$x$ + 3$y$ = 3
- 12$y$ = - 11 + 6$x$
 a. ($\frac{1}{2}$, $\frac{2}{3}$) b. (- $\frac{1}{2}$, $\frac{2}{3}$) c. ($\frac{1}{2}$, - $\frac{2}{3}$) d. ($\frac{2}{3}$, $\frac{1}{2}$)

#### Solution:

2x + 3y = 3
6x + 12y = 11
[Arrange the equations with the like terms in columns.]

- 8x - 12y = - 12
[Multiply equation 1 by - 4.]

6x + 12y = 11
______________
[Equation 2.]

- 2x           = - 1

x = 12
[Solve for x.]

2(12) + 3y = 3
[Replace x with 1 / 2 in Equation 1.]

1 + 3y = 3
[Multiply.]

y = 23
[Solve for y.]

The solution for the linear system is (1 / 2, 2 / 3).

3.
Solve the linear system.
4$x$ + $y$ = - 64
5$x$ - 4$y$ = 4
 a. (- 12, -16) b. (14, -14) c. (12, -15) d. (- 12, -15)

#### Solution:

4x + y = - 64 .....(1)
5x - 4y = 4 .....(2)

16x + 4y = - 256
[Multiply Equation 1 by 4.]

5x - 4y = 4
________________
[Equation 2.]

21x       = -252

x = - 12
[Solve for x.]

4(- 12) + y = - 64
[Substitute x = - 12 in Equation 1.]

- 48 + y = - 64
[Multiply.]

y = -16

The solution for the linear system is (- 12, -16).

4.
Solve the linear system.
$x$ - 4$y$ = 16
- 4$x$ - 7$y$ = 5
 a. (-4, -3) b. (4, -3) c. (-3, 4) d. (4, 3)

#### Solution:

x - 4y = 16 .....(1)
- 4x - 7y = 5 .....(2)

4x - 16y = 64
[Multiply Equation 1 by 4.]

- 4x - 7y = 5
______________
[Equation 2.]

-23y = 69

y = -3
[Solve for y.]

x - 4(-3) = 16
[Replace y with -3 in Equation 1.]

x + 12 = 16
[Multiply.]

x = 4
[Solve for x.]

The solution for the linear system is (4, -3).

5.
Solve the linear system.
2$x$ + 3$y$ = 25
3$x$ - 3$y$ = 0
 a. (4, 4) b. (5, 5) c. (6, 5) d. (4, 5)

#### Solution:

2x + 3y = 25 ...........(1)
3x - 3y = 0 .............(2)

2x + 3y = 25

3x -  3y = 0
____________

5x       =  25

x = 5
[Solve for x.]

2(5) + 3y = 25
[Substitute x = 5 in equation 1.]

10 + 3y = 25
[Multiply.]

3y = 15
[Subtract 10 from each side.]

y = 5
[Solve for y.]

The solution for the linear system is (5, 5).

6.
Solve the linear system.
- $x$ + $y$ = - 6
5$x$ - $y$ = 10
 a. (1, - 5) b. (5, - 5) c. (5, 1) d. (1, 5)

#### Solution:

- x + y = - 6 .........(1)
5x - y = 10 ..........(2)

- x + y = - 6

5x - y = 10
___________

4x       = 4

x = 1
[Solve for x.]

- 1 + y = - 6
[Substitute x = 1 in Equation 1.]

y = - 5
[Solve for y.]

So, the solution for the linear system is (1, - 5).

7.
Solve:
3$x$ - 4$y$ = -4
5$x$ + 4$y$ = 36
 a. (3, 5) b. (4, 5) c. (4, 4) d. (5, 3)

#### Solution:

3x - 4y = -4.............(1)
5x + 4y = 36..............(2)

3x - 4y = -4

5x + 4y = 36
______________

8x       = 32

x = 4
[Solve for x.]

3(4) - 4y = -4
[Replace x with 4 in Equation 1.]

y = 4
[Solve for y.]

The solution for the linear system is (4, 4).

8.
Solve the linear system.
- 3$x$ - $y$ = - 16
- 3$x$ + $y$ = -2
 a. (3, 8) b. (2, 7) c. (4, 7) d. (3, 7)

#### Solution:

- 3x - y = - 16..........(1)
- 3x + y = -2..........(2)

- 3x - y = - 16

- 3x + y = -2
______________

- 6x        = -18

x = 3
[Solve for x.]

- 3(3) - y = - 16
[Substitute x = 3 in Equation 1.]

y = 7
[Solve for y.]

The solution for the linear system is (3, 7).

9.
Solve the linear system.
$x$ - 2$y$ = -7
- $x$ + 7$y$ = 37
 a. (5, 6) b. (5, 9) c. (5, 5) d. (6, 5)

#### Solution:

x - 2y = -7...........(1)
- x + 7y = 37.............(2)

x - 2y = -7
[Equation 1.]

- x + 7y = 37
____________
[Equation 2.]

5y = 30

y = 6
[Solve for y.]

x - 2(6) = -7
[Substitute y = 6 in Equation 1.]

x = 5
[Solve for x.]

So, the solution for the linear system is (5, 6).

10.
Solve the linear system.
2$x$ + $y$ = 14
- 3$x$ - 2$y$ = -19
 a. (-12, 38) b. (9, - 4) c. (5, 4) d. (14, -19)

#### Solution:

2x + y = 14 .....(1)
- 3x - 2y = -19 .....(2)

Eliminate x from both the equations by making the coefficient of x in one equation opposite to the other.

6x + 3y = 42
[Multiply Equation 1 by 3.]

- 6x - 4y = -38
_______________
[Multiply Equation 2 by 2.]

-1y = 4

y = - 4
[Solve for y.]

2x + (- 4) = 14
[Substitute y = - 4 in Equation 1.]

x = 9
[Solve for x.]

The solution for the linear system is (9, - 4).