Word Problems with Systems of Equations

Setting up word problems as systems

The hardest part of word problems is translating English into equations. The strategy: identify the unknowns, find two relationships (constraints) in the problem, write an equation for each.

Step-by-step solution

Step 1 — Define variables. Let $a$ = number of apples, $o$ = number of oranges.

Step 2 — Write Equation 1 (total count). "You buy 10 fruits": $a + o = 10$

Step 3 — Write Equation 2 (total cost). "Apples cost $2 each, oranges $3, total $24": $2a + 3o = 24$

Step 4 — Solve by substitution. From Eq 1: $a = 10 - o$. Substitute into Eq 2:

$2(10 - o) + 3o = 24$ $20 - 2o + 3o = 24$ $o = 4$

Step 5 — Find $a$. $a = 10 - 4 = 6$.

Answer: 6 apples, 4 oranges.

Step 6 — Check both constraints.

• Total fruits: $6 + 4 = 10$ ✓
• Total cost: $2(6) + 3(4) = 12 + 12 = 24$ ✓

Tips for word problems

• Two unknowns need two equations. Look for two different facts in the problem.
• Common types: mixture problems, rate problems, age problems, coin problems.
• Always check your answer against the original word problem (not just the equations — you might have set up the equations wrong).

Try it in the visualization

Adjust the prices and totals. The two constraint lines are plotted on a graph — the intersection gives the answer. Try making the problem impossible (e.g., total cost too low for any combination) and see that the lines don't intersect.