Two Jobs System of Equations: Find Hours Worked

What This Problem Teaches

• Setting up systems from real constraints: Translating "total hours" and "total earnings" into mathematical equations
• Variable assignment for multiple unknowns: Choosing meaningful letters and maintaining consistency throughout
• Substitution method mastery: Expressing one variable in terms of another and making clean substitutions
• Verification habits: Checking solutions against both original constraints to build confidence
• Real-world equation modeling: Understanding how rate × time structures appear across different contexts

Visualizing the Setup

The problem gives us two constraints that must be satisfied simultaneously. We need to find values for L and C that make both equations true at the same time.

Solution: Method 1 — The Substitution Approach

Step 1 — Define variables clearly

Let L = hours worked as a lifeguard
Let C = hours worked as a cashier

Step 2 — Write the constraint equations

From the total hours worked: L + C = 14

From the total earnings: 8L + 6C = 96

The first equation captures the time constraint, while the second captures the earnings constraint. Both must be satisfied.

Step 3 — Solve the first equation for one variable

From L + C = 14, we can express C in terms of L:

Step 4 — Substitute into the earnings equation

Replace C with (14 - L) in the earnings equation:

Step 5 — Solve for L

Distribute the 6:

Combine like terms:

Subtract 84 from both sides:

Divide by 2:

Step 6 — Find C using the substitution

Using C = 14 - L:

Solution: Method 2 — The Elimination Strategy

Step 1 — Set up the system

Starting with our two equations:

Step 2 — Eliminate one variable by multiplication

To eliminate C, multiply Equation 1 by -6:

Step 3 — Add the equations to eliminate C

Add the modified equations vertically:

Step 4 — Substitute back to find C

Using L = 6 in Equation 1:

Verification

Let's confirm our solution satisfies both original constraints:

Check total hours

L + C = 6 + 8 = 14 ✓

Check total earnings

8L + 6C = 8(6) + 6(8) = 48 + 48 = 96 ✓

Both constraints are satisfied, confirming our solution is correct.

Common Pitfalls

The Pattern Behind This

This problem follows the classic mixture with two components structure that appears throughout mathematics:

Whether you're mixing coffee beans at different prices, combining investments with different returns, or blending solutions with different concentrations, the mathematical structure remains the same. The "rates" change (price per pound, interest rate, concentration percentage), but the two-equation setup is identical.

This pattern is so robust that once you recognize it, you can solve these problems almost automatically — define variables for the two components, write one equation for total quantity, and one for total value.

Why This Shows Up Everywhere

• Payroll management: Companies tracking employee hours across different pay scales or departments
• Project budgeting: Allocating resources between high-cost specialists and lower-cost general staff
• Manufacturing optimization: Using machines with different operating costs to meet production targets
• Portfolio analysis: Balancing investments between assets with different risk-return profiles

The mathematical thinking you're developing here — setting up constraints and solving systems — is fundamental to operations research, economics, and any field where resources must be allocated efficiently.

What If?