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Difficulty: MediumLinear Equations in One Variable

A manufacturing plant operates two assembly lines, Line X and Line Y. Line X produces 15 units per hour, and Line Y produces 22 units per hour. On a certain day, Line X operated for 3 hours longer than Line Y did, and the two lines produced a total of 415 units. How many hours did Line Y operate?

Answer: 10 hours

Answer

10
Let hh represent the number of hours Line Y operated. Because Line X operated for 3 hours longer than Line Y, Line X operated for h+3h + 3 hours. The total number of units produced by both lines is the sum of their individual outputs: 15(h+3)+22h=41515(h + 3) + 22h = 415. Distributing 15 yields 15h+45+22h=41515h + 45 + 22h = 415. Combining like terms gives 37h+45=41537h + 45 = 415. Subtracting 45 from both sides yields 37h=37037h = 370, and dividing by 37 gives h=10h = 10. Therefore, Line Y operated for 10 hours.

Step-by-Step Solution

1
Define the variable representing Line Y's operating time in hours.
Let hh represent the number of hours Line Y operated. Line X's operating time is h+3h + 3 hours.
Line X operated for 3 hours longer than Line Y.
2
Formulate a linear equation in one variable for total units produced.
15(h+3)+22h=41515(h + 3) + 22h = 415
Total production is the sum of production from Line X (15×(h+3)15 \times (h + 3)) and Line Y (22×h22 \times h).
3
Distribute and combine like terms on the left side of the equation.
15h+45+22h=415  ⟹  37h+45=41515h + 45 + 22h = 415 \implies 37h + 45 = 415
Apply the distributive property and combine variable terms.
4
Isolate the variable hh.
37h=370  ⟹  h=1037h = 370 \implies h = 10
Subtract 45 from both sides of the equation and divide by 37.

Key Concept

Linear Equations in One Variable
Estimated Time:1m 30s
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Topics

Linear Equations in One Variable

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