Math Problem Statement

solve

Two mechanics worked on a car. The first mechanic charged $45 per hour, and the second mechanic charged $80 per hour. The mechanics worked for a combined total of 20 hours, and together they charged a total of $1425. How long did each mechanic work?

Solution

Let's solve the problem step by step.

Problem Summary:

  • The first mechanic charges $45/hour.
  • The second mechanic charges $80/hour.
  • Combined, they worked for 20 hours.
  • They earned a total of $1,425.
  • Find how many hours each mechanic worked.

Define Variables:

Let:

  • xx = hours worked by the first mechanic.
  • yy = hours worked by the second mechanic.

Equations:

  1. Total hours: x+y=20x + y = 20
  2. Total earnings: 45x+80y=142545x + 80y = 1425

Solve the System of Equations:

From Equation (1):

y=20xy = 20 - x

Substitute y=20xy = 20 - x into Equation (2):

45x+80(20x)=142545x + 80(20 - x) = 1425 45x+160080x=142545x + 1600 - 80x = 1425 35x+1600=1425-35x + 1600 = 1425 35x=175-35x = -175 x=5x = 5

Substitute x=5x = 5 into Equation (1):

5+y=205 + y = 20 y=15y = 15

Solution:

  • The first mechanic worked 5 hours.
  • The second mechanic worked 15 hours.

Would you like a detailed breakdown or have additional questions? Here are some related questions to explore further:

  1. What if the hourly rates were swapped? How would the solution change?
  2. How would you graphically represent this system of equations?
  3. What happens if the total hours worked were different but earnings stayed the same?
  4. Can you verify the solution by checking total earnings for the two mechanics?
  5. How can this system of equations be solved using matrices?

Tip: Always double-check your solution by substituting values back into the original equations!

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Linear Equations

Formulas

x + y = Total Hours
45x + 80y = Total Earnings

Theorems

Substitution Method for Solving Systems of Equations

Suitable Grade Level

Grades 8-10