Solved Examples and Worksheet for Word Problems on Systems of Equations

Q1A 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
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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
Q2A mechanical plant hires 790 labors on a daily wage scheme paying $6,432. Men are paid $10 and women are paid $6. Find the number of men and women hired.

A. 368 men, 422 women
B. 367 men, 423 women
C. 423 men, 367 women
D. 422 men, 368 women

Solutions
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Step: 1
Let number of men be x.
Step: 2
Let number of women be y.
Step: 3
x + y = 790 --- (1)
  [Linear equation for the number of men and women hired.]
Step: 4
10x + 6y = 6432 --- (2)
  [Equation for the daily wages paid.]
Step: 5
y = 790 - x
  [Revise equation 1.]
Step: 6
10x + 6(790 - x) = 6432
  [Substitute y = 790 - x in Equation 2.]
Step: 7
4x + 4740 = 6432
  [Combine like terms.]
Step: 8
4x = 1692
  [Subtract 4740 from each side.]
Step: 9
x = 423
  [Divide each side by 4.]
Step: 10
y = 790 - 423 = 367
  [Substitute x = 423 in revised Equation 3.]
Step: 11
367 women and 423 men are hired.
Correct Answer is :   423 men, 367 women
Q3One pound of sugar costs $5, and one pound of flour costs $4. Carol bought 19 quarts in all, and she paid $23 more for sugar than for flour. How many quarts of sugar and flour did she buy?
A. Sugar: 11, Flour 11
B. Sugar: 8, Flour 11
C. Sugar: 8, Flour 8
D. Sugar: 11, Flour 8

Solutions
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Step: 1
Let x be the number of pound of sugar and y be the number of pound of flour bought by Carol.
Step: 2
x + y = 19 - - - - - - - (1)
  [As per the question.]
Step: 3
5x - 4y = 23 - - - - - - - - - (2)
  [As per the question.]
Step: 4
4x + 4y = 76
  [Multiply the first equation by 4.]
Step: 5
9x = 99
  [Solve the equations in step 3 and step 4.]
Step: 6
x = 11
  [Divide both sides by 9.]
Step: 7
y = 19 - x
Step: 8
y = 8
  [Replace x = 11.]
Step: 9
Therefore, Carol bought 11 pound of sugar and 8 pound of flour
Correct Answer is :   Sugar: 11, Flour 8
Q4Tim bought 5 books and a pen for $79 and Jerald bought a book and 16 pens of the same kind for $158. Find the prices of the book and the pen.

A. $79, $158
B. $13, $9
C. $9, $23
D. $14, $9

Solutions
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Step: 1
Let the price of the book = $x
Step: 2
Let the price of the pen = $y
Step: 3
5x + y = 79
  [First equation from the data.]
Step: 4
x + 16y = 158
  [Second equation from the data.]
Step: 5
y = 79 - 5x
  [Solve the first equation for y.]
Step: 6
x + 16(79 - 5x) = 158
  [Substitute the values.]
Step: 7
x - 80x + 1264 = 158
Step: 8
- 79x = 158 - 1264
Step: 9
- 79x = - 1106
  [Subtract.]
Step: 10
x = 14
  [Divide throughout by - 79.]
Step: 11
16y = 158 - x = 158 - 14 = 144
  [Substitute the values.]
Step: 12
y = 9
  [Divide throughout by 16.]
Step: 13
So, the price of the book is $14 and the price of the pen is $9.
Correct Answer is :   $14, $9
Q5Paula purchased a total of 47 books and toys for the Taloga play school. Each book costs $23 and each toy costs $12. How many books and toys did she buy for $839?
A. 24 books and 23 toys
B. 25 books and 22 toys
C. 23 books and 24 toys
D. 22 books and 25 toys

Solutions
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Step: 1
Let x be the number of books and y be the number of toys, Paula purchased.
Step: 2
x + y = 47 --- (1)
  [Express as a linear equation.]
Step: 3
23x + 12y = 839 --- (2)
  [Equation for the total cost of the books and toys.]
Step: 4
23x + 12(- x + 47) = 839
  [From equation 1, y = - x + 47. Substitute it in equation 2.]
Step: 5
11x + 564 = 839
  [Group the like terms.]
Step: 6
11x = 275
  [Subtract 564 from the two sides of the equation.]
Step: 7
x = 25
  [Divide throughout by 11.]
Step: 8
y = - (25) + 47 = 22
  [Substitute the values.]
Step: 9
Paula bought 25 books and 22 toys.
Correct Answer is :   25 books and 22 toys
Q6Gary bought 10 candies and cookies for $24. The cost of a candy is $1 and the cost of a cookie is $0.9. Let x be the number of candies and y be the number of cookies. Which system of linear equations represents the situation?
A. x + y = 10 and x + 0.9y = 24
B. x + y = 10 and 0.9x + y = 24
C. x + y = 24 and 0.9x + y = 10
D. x + y = 24 and x + 0.9y = 10

Solutions
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Step: 1
x + y = 10
  [Total number of candies and cookies = 10.]
Step: 2
Cost of x candies is $x.
  [Cost of a candy = $1.]
Step: 3
Cost of y cookies is $0.9y.
  [Cost of a cookie = $0.9.]
Step: 4
Cost of 10 candies and cookies is $24, so x + 0.9y = 24
  [Total cost = $24.]
Step: 5
So, the system of equations are x + y = 10 and x + 0.9y = 24.
Correct Answer is :   x + y = 10 and x + 0.9y = 24
Q7Josh bought 28 soaps and shampoo bottles for $174. Cost of one soap is $3and cost of one shampoo bottle is $10. Let x be the number of soaps and y be the number of shampoo bottles. Identify the linear system of equations that represent the situation.
A. x + y = 28 and 10x + 3y = 174
B. x + y = 28 and 3x + 10y = 174
C. x + y = 174 and 10x + 3y = 28
D. x + y = 174 and 3x + 10y = 28

Solutions
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Step: 1
x + y = 28
  [Total number of soaps and shampoo bottles is 28.]
Step: 2
Cost of x soaps = $3x
  [Cost of a soap = $3.]
Step: 3
Cost of y shampoo bottles = $10y
  [Cost of a shampoo bottle is $10.]
Step: 4
Cost of 28 soaps and shampoo bottles is $174. So, 3x + 10y = 174
Step: 5
So, the system of equations are x + y = 28 and 3x + 10y = 174.
Correct Answer is :   x + y = 28 and 3x + 10y = 174
Q8Which of the following would be a good first step to solve the linear system?
- x + y = - 2
3x - y = - 4

A. Substitute x = -2 - y for x in the second equation
B. Substitute y = -4 - x for y in the first equation
C. Substitute x = - 2 - y for y in the second equation
D. Adding the two equations.

Solutions
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Step: 1
- x + y = - 2
  [Equation 1.]
Step: 2
3x - y = - 4
____________
  [Equation 2.]
Step: 3
2x      = - 6
  [Add Equation 1 and Equation 2.]
Step: 4
So, adding the two equations is a good step to solve the linear system of equations.
Correct Answer is :   Adding the two equations.
Q9Identify the first step to solve the linear system.
- x + y = 6      
5x - 4y = -12  
A. Substitute y = 6 + x for x in the second equation.
B. Substitute x = 6 + y for x in the second equation.
C. Substitute y = 6 - x for x in the second equation.
D. Substitute y = 6 + x for y in the second equation.

Solutions
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Step: 1
- x + y = 6
  [Equation 1.]
Step: 2
5x - 4y = -12
  [Equation 2.]
Step: 3
So, substitute y = 6 + x for y in the second equation is the first step to solve the system of equations.
Correct Answer is :   Substitute y = 6 + x for y in the second equation.
Q10If 11 times the larger of the two numbers is divided by the smaller one, we get 6 as quotient and 14 as the remainder. Also if 10 times the smaller number is divided by the larger one, we get 4 as quotient and 34 as remainder. Find the numbers.

A. 4, 7
B. 5, 4
C. 5, 6
D. 7, 4

Solutions
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Step: 1
Let x be the smaller number and y be the larger number.
Step: 2
11y = 6x + 14 - - - - - - - - - - - (1) and 10x = 4y + 34 - - - - - - - - - - - - -(2)
  [As per the question.]
Step: 3
- 24x + 44y = 56
  [Multiply equation (1) with 4.]
Step: 4
110x - 44y = 374
  [Multiply equation (2) with 11.]
Step: 5
86x = 430
  [Add.]
Step: 6
x = 5
  [Divide both sides by 86.]
Step: 7
11y = 6(5) + 14
  [Replace x with 5 in equation (1).]
Step: 8
11y = 44
Step: 9
y = 4
Step: 10
So, the two numbers are 5 and 4.
Correct Answer is :   5, 4

HighSchool Math