﻿ Solving Linear Systems (using Linear Combinations) Worksheet | Problems & Solutions

# Solving Linear Systems (using Linear Combinations) Worksheet

Solving Linear Systems (using Linear Combinations) Worksheet
• Page 1
1.
Solve the linear system.
4$x$ + 5$y$ = 45
7$x$ - 5$y$ = 10
 a. (6, 5) b. (4, 5) c. (5, 5) d. (4, 4)

#### Solution:

4x + 5y = 45
[Equation 1.]

7x -  5y = 10
____________
[Equation 2.]

11x       =  55

x = 5
[Solve for x.]

4(5) + 5y = 45
[Substitute x = 5 in equation 1.]

20 + 5y = 45
[Multiply.]

5y = 25
[Subtract 20 from each side.]

y = 5
[Solve for y.]

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

2.
Solve the linear system.
-$x$ + $y$ = -20
17$x$ - $y$ = 68
 a. (17, 3) b. (3, -17) c. (17, -17) d. None of the above

#### Solution:

-x + y = -20
[Equation 1.]

17x - y = 68
___________
[Equation 2.]

16x       = 48

x = 3
[Solve for x.]

-3 + y = -20
[Substitute x = 3 in Equation 1.]

y = -17
[Solve for y.]

So, the solution for the linear system is (3, -17).

3.
Which of the following ordered pairs satisfies the linear system?
3$x$ - 4$y$ = 4
5$x$ + 4$y$ = 28
 a. (5, 1) b. (4, 2) c. (4, 3) d. (3, 3)

#### Solution:

3x - 4y = 4
[Equation 1.]

5x + 4y = 28
______________
[Equation 2.]

8x       = 32

x = 4
[Solve for x.]

3(4) - 4y = 4
[Substitute 4 for x in equation 1.]

y = 2
[Solve for y.]

The solution for the linear system is (4, 2).

4.
Solve the linear system.
-5$x$ - $y$ = -15
-3$x$ + $y$ = -1
 a. (1, 5) b. (2, 5) c. (2, 6) d. None of the above

#### Solution:

-5x - y = -15
[Equation 1.]

-3x + y = -1
______________
[Equation 2.]

-8x        = -16

x = 2
[Solve for x.]

-5(2) - y = -15
[Substitute x = 2 in Equation 1.]

y = 5
[Solve for y.]

The solution for the linear system is (2, 5).

5.
Solve the linear system.
$x$ - 3$y$ = -13
-$x$ + 7$y$ = 37
 a. (5, 6) b. (6, 5) c. (5, 9) d. None of the above

#### Solution:

x - 3y = -13
[Equation 1.]

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

4y = 24

y = 6
[Solve for y.]

x - 3(6) = -13
[Substitute y = 6 in Equation 1.]

x = 5
[Solve for x.]

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

6.
Solve the linear system.
3$x$ + $y$ = 22
-4$x$ - 2$y$ = -26
 a. (-9, 5) b. (9, 5) c. (9, -5) d. (-9, -5)

#### Solution:

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

12x + 4y = 88
[Multiply Equation 1 by 4.]

-12x - 6y = -78
_______________
[Multiply Equation 2 by 3.]

-2y = 10

y = -5
[Solve for y.]

3x + (-5) = 22
[Substitute y = -5 in Equation 1.]

x = 9
[Solve for x.]

The solution for the linear system is (9, -5).

7.
Which of the following ordered pairs satisfies the given linear system?
$x$ - 4$y$ = -22
-3$x$ - 5$y$ = -36
 a. (-2, 6) b. (-2, 0) c. (2, 6) d. None of the above

#### Solution:

3x - 12y = -66
[Multiply Equation 1 by 3.]

-3x - 5y = -36
______________
[Equation 2.]

-17y = -102

y = 6
[Solve for y.]

x - 4(6) = -22
[Substitute y = 6 in Equation 1.]

x - 24 = -22
[Multiply.]

x = -22 + 24

x = 2
[Solve for x.]

So, the solution for the linear system is (2, 6).

8.
Which of the following ordered pairs satisfies the given linear system?
-$x$ + $y$ = -2 --- Equation 1
-3$x$ - 3$y$ = -30 --- Equation 2
 a. (6, -4) b. (6, 4) c. (-6, 4) d. None of the above

#### Solution:

-x - y = -10
[Divide Equation 2 by 3.]

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

-x - y = -10
_____________
[Revised Equation 2.]

-2x      = -12
[Add Equation 1 and revised Equation 2.]

x = 6
[Solve for x.]

-6 + y = -2
[Substitute 6 for x in revised Equation 2.]

y = 4

The solution for the linear system is (6, 4).

9.
Which of the following solutions satisfies the given linear system?
5$x$ - $y$ = -2
-4$x$ + 5$y$ = -11
 a. (-2, -3) b. (-3, 1) c. (-1, -3) d. None of the above

#### Solution:

25x - 5y = -10
[Multiply Equation 1 by 5.]

-4x + 5y = -11
______________
[Multiply Equation 2 by 1.]

21x         = -21

x = -1
[Solve for x.]

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

y = -3
[Solve for y.]

Solution for the linear system is (-1, -3).

10.
Which of the following ordered pairs satisfies the given linear system?
-5$x$ + 4$y$ = -4
4$y$ = 4 + 3$x$
 a. (4, 4) b. (-4, 5) c. (3, 4) d. None of the above

#### Solution:

-5x + 4y = -4
[Equation 1.]

3x - 4y = -4
____________
[Rearrange Equation 2 and multiply with -1.]

-2x       = -8

x = 4
[Solve for x.]

-5(4) + 4y = -4
[Substitute 4 for x in Equation 1.]

-20 + 4y = -4
[Multiply.]

4y = 16