Solved Examples and Worksheet for Solving Systems of Equations by Substitution Method

Q1The price of each marble was $6 and the price of each dice was $4. A total of 82 marbles and dice were purchased for $424. Find the number of marbles and the number of dice purchased.

A. 48 marbles, 34 dice
B. 50 marbles, 32 dice
C. 46 marbles, 36 dice
D. 44 marbles, 38 dice

Solutions
Share 
Step: 1
Let x be the number of marbles and y be the number of dice purchased.
Step: 2
x + y = 82
  [Equation 1.]
Step: 3
6x + 4y = 424
  [Equation 2.]
Step: 4
x = 82 - y
  [Rearrange Equation 1.]
Step: 5
6(82 - y) + 4y = 424
  [Substitute x = 82 - y in Equation 2.]
Step: 6
-2y + 492 = 424
  [Combine like terms.]
Step: 7
-2y = -68
  [Subtract 492 from each side.]
Step: 8
y = 34
  [Divide each side by -2.]
Step: 9
x = 82 - 34 = 48
  [Substitute y = 34 in the revised equation 1.]
Step: 10
48 marbles and 34 dice are purchased.
Correct Answer is :   48 marbles, 34 dice
Q2Which of the following linear systems has the solution (3, 0)?

A. x + 3y = -3 and x + 4y = -3
B. x - 3y = -3 and x + 4y = 3
C. -x + 4y = 3 and x - 4y = 3
D. x + 4y = 3 and x - 4y = 3

Solutions
Share 
Step: 1
Substitute (3, 0) in all the linear systems and verify.
Step: 2
For the linear system x + 4y = 3, x - 4y = 3:
3 + 4(0) = 3 + 0 = 3
  [Substitute (3, 0) in x + 4y = 3.]
Step: 3
3 - 4(0) = 3 - 0 = 3
  [Substitute (3, 0) in x - 4y = 3. ]
Step: 4
So, the linear system in the choice D has the solution (3, 0).
Correct Answer is :   x + 4y = 3 and x - 4y = 3
Q3Which of the following ordered pairs satisfies the linear system?
4x - 3y = -6   [Equation 1.]
-x + 4y = -5 [Equation 2.]

A. (-4, 2)
B. (-3, 2)
C. (-4, -2)
D. (-3, -2)

Solutions
Share 
Step: 1
-x + 4y = -5
  [Original equation 2.]
Step: 2
x = 5 + 4y
  [Revise equation 2.]
Step: 3
4(5 + 4y) - 3y = -6
  [Substitute 5 + 4y for x in equation 1.]
Step: 4
13y + 20 = -6
  [Combine like terms.]
Step: 5
13y = -26
  [Subtract 20 from each side.]
Step: 6
y = -2
  [Divide each side by 13.]
Step: 7
x = 5 + 4y = 5 + 4(-2)
  [Substitute -2 for y in the revised equation 2.]
Step: 8
x = -3
  [Simplify.]
Step: 9
The solution for the linear system is (-3, -2).
Correct Answer is :   (-3, -2)
Q4Diane purchased a total of 25 books and toys for the Taloga play school. Each book cost $17 and each toy cost $9. How many books and toys did she buy for $345?
A. 10 books and 15 toys
B. 14 books and 11 toys
C. 11 books and 14 toys
D. 15 books and 10 toys

Solutions
Share 
Step: 1
Let x be the number of books and y be the number of toys, Diane purchased.
Step: 2
x + y = 25 --- (1)
  [Linear equation for the total books and toys.]
Step: 3
17x + 9y = 345 --- (2)
  [Equation for the total cost of the books and toys.]
Step: 4
17x + 9(-x + 25) = 345
  [From equation 1, y = -x + 25. Substitute it in equation 2.]
Step: 5
8x + 225 = 345
  [Combine like terms.]
Step: 6
8x = 120
  [Subtract 225 from each side.]
Step: 7
x = 15
  [Divide each side by 8.]
Step: 8
y = -(15) + 25 = 10
  [Substitute x = 15 in equation 1.]
Step: 9
Diane bought 15 books and 10 toys.
Correct Answer is :   15 books and 10 toys
Q5Find the ordered pair, which is the solution of the linear system.
-4x + 3y = 11
-5x + 2y = 19

A. (5, -3)
B. (-5, 3)
C. (5, 3)
D. (-5, -3)

Solutions
Share 
Step: 1
Substitute all the ordered pairs in the equations and verify which ordered pair satisfies both the equations.
Step: 2
For the ordered pair (-5, -3):
Step: 3
-4x + 3y = 11
  [Equation 1.]
Step: 4
-4(-5) + 3(-3) = 11
  [Substitute x = 5 and y = -3.]
Step: 5
11 = 11
  [Simplify.]
Step: 6
-5x + 2y = 19
  [Equation 2.]
Step: 7
-5(-5) + 2(-3) = 19
  [Substitute x = -5 and y = -3.]
Step: 8
19 = 19
  [Simplify.]
Step: 9
So, (-5, -3) is the solution of the linear system.
Correct Answer is :   (-5, -3)
Q6Find the ordered pair, which is the solution of the linear system.
9x - 2y = -22
-9x + 4y = 80

A. (4, -29)
B. (-4, -29)
C. (4, 29)
D. (-4, 29)

Solutions
Share 
Step: 1
Substitute all the ordered pairs in the equations and verify which ordered pair satisfies both the equations.
Step: 2
For the ordered pair (4, 29):
Step: 3
9x - 2y = -22
  [Equation 1.]
Step: 4
9(4) - 2(29) = -22
  [Substitute x = 4 and y = 29.]
Step: 5
-22 = -22
  [Simplify.]
Step: 6
-9x + 4y = 80
  [Equation 2.]
Step: 7
-9(4) + 4(29) = 80
  [Substitute x = 4 and y = 29.]
Step: 8
80 = 80
  [Simplify.]
Step: 9
So, (4, 29) is the solution of the linear system.
Correct Answer is :   (4, 29)
Q7Find the ordered pair, which is the solution of the linear system.
-5x + 5y = 0
8x - 7y = -4
A. (-4, 4)
B. (4, 4 )
C. (4 , -4 )
D. (-4 , - 4)

Solutions
Share 
Step: 1
Substitute all the ordered pairs in the equations and verify which ordered pair satisfies both the equations.
Step: 2
For the ordered pair (-4, -4):
Step: 3
-5x + 5y = 0
  [Equation 1.]
Step: 4
-5(-4) + 5(-4) = 0
  [Substitute x = -4 and y = -4 in equation 1.]
Step: 5
0 = 0
  [Simplify.]
Step: 6
8x - 7y = -4
  [Equation 2.]
Step: 7
8(-4) - 7(-4) = -4
  [Substitute x = -4 and y = -4 in equation 2.]
Step: 8
-4 = -4
  [Simplify.]
Step: 9
So, (-4, -4) is the solution of the linear system.
Correct Answer is :   (-4 , - 4)
Q8Which of the following ordered pairs satisfies the linear system?
a = b - 5       [Equation 1.]
3a + b = 25   [Equation 2.]

A. (-5, 10)
B. (-5, -10)
C. (5, 10)
D. (5, -10)

Solutions
Share 
Step: 1
a = b - 5
  [Original equation 1.]
Step: 2
3(b - 5) + b = 25
  [Substitute b - 5 for a in equation 2.]
Step: 3
4b - 15 = 25
  [Combine like terms.]
Step: 4
4b = 40
  [Add 15 to each side.]
Step: 5
b = 10
  [Divide each side by 4.]
Step: 6
a = b - 5 = 10 - 5
  [Substitute 10 for b in equation 1.]
Step: 7
a = 5
  [Simplify.]
Step: 8
The solution for the linear system is (5, 10).
Correct Answer is :   (5, 10)
Q9Which of the following ordered pairs satisfies the linear system?
-p + q = 7      [Equation 1]
3p + 5q = 11   [Equation 2]
A. (-3, -4)
B. (3, -4)
C. (3, 4)
D. (-3, 4)

Solutions
Share 
Step: 1
q = p + 7
  [Rearrange Equation 1.]
Step: 2
3p + 5(p + 7) = 11
  [Substitute p + 7 for q in Equation 2.]
Step: 3
8p + 35 = 11
  [Combine like terms.]
Step: 4
8p = -24
  [Subtract 35 from each side.]
Step: 5
p = -3
  [Divide each side by 8.]
Step: 6
q = p + 7 = -3 + 7
  [Substitute -3 for p in revised Equation 1.]
Step: 7
q = 4
  [Simplify.]
Step: 8
The solution for the linear system is (-3, 4).
Correct Answer is :   (-3, 4)
Q10Solve the linear system.
w + z = 3
4w = 12
A. w = 0, z = -3
B. w = -3, z = 0
C. w = 0, z = 3
D. w = 3, z = 0

Solutions
Share 
Step: 1
w + z = 3
  [Equation 1.]
Step: 2
4w = 12
  [Equation 2.]
Step: 3
w = 3
  [Divide each side by 4.]
Step: 4
3 + z = 3
  [Substitute w = 3 in the Equation 1.]
Step: 5
z = 0
  [Subtract 3 from each side.]
Step: 6
The solution for the linear system is (3, 0).
Correct Answer is :   w = 3, z = 0
Q11Which of the following ordered pairs satisfies the linear system? s - t = 0      [Equation 1.]
4s - 9t = -10 [Equation 2.]

A. (-2, 2)
B. (2, -2)
C. (2, 2)
D. (-2, -2)

Solutions
Share 
Step: 1
s = t
  [Revise equation 1.]
Step: 2
4s - 9s = -10
  [Substitute s for t in equation 2.]
Step: 3
-5s = -10
  [Combine like terms.]
Step: 4
s = 2
  [Divide each side by -5.]
Step: 5
t = 2
  [From revised Equation 1.]
Step: 6
The solution for the linear system is (2, 2).
Correct Answer is :   (2, 2)
Q12A circus company sold 209 tickets on a particular day. The entry fee was $13 for an adult and $3 for a child. The total amount collected was $1477. How many adults and children went to the circus that day?
A. 85 adults and 124 children
B. 84 adults and 125 children
C. 125 adults and 84 children
D. 124 adults and 85 children

Solutions
Share 
Step: 1
Let x be the number of adults.
Step: 2
Let y be the number of children.
Step: 3
Total number of tickets = 209
Step: 4
So, Number of adults + Number of children = 209
Step: 5
x + y = 209
  [Equation 1.]
Step: 6
Total amount collected =$1477
Step: 7
So, (Cost of an adult ticket) x (Number of adults) + (Cost of a child ticket) x (Number of children) = $1477
Step: 8
13x + 3y = 1477
  [Substitute.]
Step: 9
13(-y + 209) + 3y = 1477
  [Substitute -y + 209 for x in Equation 2.]
Step: 10
-10y + 2717 = 1477
  [Combine like terms.]
Step: 11
-10y = -1240
  [Subtract 2717 from each side.]
Step: 12
y = 124
  [Divide each side by -10.]
Step: 13
x = -(124) + 209 = 85
  [Substitute 124 for y in revised Equation 1.]
Step: 14
85 adults and 124 children went to the circus that day.
Correct Answer is :   85 adults and 124 children
Q13An Opera house collected $3560 by selling the tickets. The people sitting in the row A are charged $16 and the people sitting in row B are charged $12. The total number of tickets sold is 250. Find the number of people sitting in row A and row B.
A. 111 in row A 139 in row B
B. 139 in row A 111 in row B
C. 140 in row A 110 in row B
D. 110 in row A 140 in row B

Solutions
Share 
Step: 1
Number of people in row A = x
Step: 2
Number of people in row B = y
Step: 3
x + y = 250 --- (1)
  [Linear Equation for the people sitting in row A and row B.]
Step: 4
16x + 12y = 3560 --- (2)
  [Equation for the money collected by selling tickets.]
Step: 5
y = 250 - x
  [Revised Equation 1.]
Step: 6
16x + 12(250 - x) = 3560
  [Substitute y = 250 - x for y in Equation 1.]
Step: 7
4x + 3000 = 3560
  [Combine like terms.]
Step: 8
4x = 560
  [Subtract 3000 from both sides.]
Step: 9
x = 140
  [Divide each side by 4.]
Step: 10
y = 250 - 140 = 110
  [Substitute x = 140 in the revised Equation 1.]
Step: 11
The number of people sitting in row A are 140 and the number of people sitting in the row B are 110.
Correct Answer is :   140 in row A 110 in row B
Q14A mechanical plant hires 770 labors on a daily wage scheme paying $6244. Men are paid $9 and women are paid $7. Find the number of men and women hired.
A. 427 men, 343 women
B. 426 men, 344 women
C. 343 men, 427 women
D. 344 men, 426 women

Solutions
Share 
Step: 1
Let number of men be x.
Step: 2
Let number of women be y.
Step: 3
x + y = 770 --- (1)
  [Linear equation for the number of men and women hired.]
Step: 4
9x + 7y = 6244 --- (2)
  [Equation for the daily wages paid.]
Step: 5
y = 770 - x
  [Revise equation 1.]
Step: 6
9x + 7(770 - x) = 6244
  [Substitute y = 770 - x in Equation 2.]
Step: 7
2x + 5390 = 6244
  [Combine like terms.]
Step: 8
2x = 854
  [Subtract 5390 from each side.]
Step: 9
x = 427
  [Divide each side by 2.]
Step: 10
y = 770 - 427 = 343
  [Substitute x = 427 in revised Equation 3.]
Step: 11
343 women and 427 men are hired.
Correct Answer is :   427 men, 343 women
Q15Which of the following linear systems has the solution (2, 0)?

A. x + 2y = - 2 and x + 3y = - 2
B. x - 2y = - 2 and x + 3y = 2
C. - x + 2y = 2 and x - 3y = 2
D. x + 2y = 2 and x - 3y = 2

Solutions
Share 
Step: 1
Substitute (2, 0) in all the linear systems and verify.
Step: 2
For the linear system x + 2y = 2, x - 3y = 2:
2 + 2(0) = 2 + 0 = 2
  [Substitute (2, 0) in x + 2y = 2.]
Step: 3
2 - 3(0) = 2 - 0 = 2
  [Substitute (2, 0) in x - 3y = 2. ]
Step: 4
So, the linear system in the choice D has the solution (2, 0).
Correct Answer is :   x + 2y = 2 and x - 3y = 2

HighSchool Math