#### 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 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

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 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, $9 Step: 1 Let the price of the book =$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

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 $0.9. Let x be the number of candies and y be the number of cookies. Which system of linear equations represents the situation? A. x + y = 10 and x + 0.9y = 24 B. x + y = 10 and 0.9x + y = 24 C. x + y = 24 and 0.9x + y = 10 D. x + y = 24 and x + 0.9y = 10 Step: 1 x + y = 10 [Total number of candies and cookies = 10.] Step: 2 Cost of x candies is$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

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

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.
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.

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

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