Solve the Fruit Stand System of Equations

What This Problem Teaches

Visualizing the Problem

Let's organize what we know in a clear table format:

Here, a represents the price of one apple and o represents the price of one orange.

Solution: Method 1 — Elimination by Scaling

Step 1 — Define variables and write the system

Let a = price of one apple (in dollars)
Let o = price of one orange (in dollars)

Step 2 — Scale Equation 1 to match coefficients

Notice that Steve bought exactly twice as many oranges as Sally (10 vs 5). If we multiply Sally's entire purchase by 2, we'll have matching orange coefficients:

Step 3 — Eliminate the orange variable

Now both Equation 2 and Equation 3 have 10o. Subtract Equation 3 from Equation 2:

Step 4 — Solve for the orange price

Substitute a = 2.25 into Equation 1:

Solution: Method 2 — Purchase Comparison Strategy

Here's a different way to think about the problem. Instead of formal elimination, let's compare what the two shoppers bought.

Step 1 — Analyze the purchase difference

Steve bought 4 more apples (7 - 3) and 5 more oranges (10 - 5) than Sally, and paid $15.50 more ($28.75 - $13.25):

Step 2 — Compare with Sally's scaled purchase

From Sally's purchase, we know 3a + 5o = 13.25. The difference between Steve's extras and Sally's purchase is:

Step 3 — Find the orange price

Using a = 2.25 in Sally's equation:

Verification

Let's check both prices in both original equations:

Sally's purchase check:
3(2.25) + 5(1.30) = 6.75 + 6.50 = 13.25 ✓

Steve's purchase check:
7(2.25) + 10(1.30) = 15.75 + 13.00 = 28.75 ✓

Both equations are satisfied, confirming our solution is correct.

Watch Out For These

The Pattern Behind This

This problem follows the standard structure for 2×2 systems of linear equations. The general pattern is:

The key insight is recognizing when one purchase is a clean multiple of another's quantities. Here, Steve bought exactly 2× Sally's oranges, making the scaling obvious. In general, look for the least common multiple of the coefficients you want to eliminate.

This technique works whenever you have two linear equations in two unknowns, whether they represent fruit prices, travel times, or investment returns. The elimination method is particularly powerful when the coefficients have simple ratios.

How to Spot This Problem Type

Systems of equations problems typically include these tell-tale phrases:

• "Two people buy..." with different quantities and totals
• "Find the cost of each item" when multiple item types are mixed
• "Write two equations" explicitly asking for system setup
• Two scenarios with the same unknowns but different numbers

Any time you see two separate purchases (or transactions, or scenarios) involving the same unknown prices or quantities, you're looking at a system of equations. The word "and" often signals that two unknowns need to be found simultaneously.

Four "What-If?" Problems