Systems Word Problems
Practice building and solving two-equation models from cost, mixture, and age scenarios.
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Questions Covered in This Set
10 cards to master
What are the 4 steps for a systems word problem?
1) Name the variables explicitly, 2) find the two facts, 3) solve by substitution or elimination, 4) re-read the question to answer what was asked.
What two kinds of equations do most word problems give?
A count equation (how many things) and a value equation (how much money, mass, points, or acid).
Set up: 200 tickets sold for $1,700; adults $12, children $5.
a + c = 200 and 12a + 5c = 1700, giving a = 100 and c = 100.
Mixture setup: 20% and 50% acid mixed to make 60 mL of 30%.
x + y = 60 and 0.20x + 0.50y = 0.30(60) = 18, so x = 40 mL and y = 20 mL.
How does concentration intuition help check a mixture answer?
The mixture uses more of the solution whose concentration is closest to the target (30% is nearer 20%, so more of the 20% solution).
Age setup: Maria is 3 times her brother's age; in 8 years she'll be twice his age.
m = 3b and m + 8 = 2(b + 8), so b = 8 and Maria is 24.
What is the classic mistake in age problems?
Forgetting to add the time shift to BOTH people's ages.
Shortcut when asked only for x + y: 3x + 2y = 19 and 2x + 3y = 16.
Add the equations: 5x + 5y = 35, so x + y = 7 — no need to find x and y separately.
When should you use substitution vs. elimination?
Substitute if a variable is already isolated; eliminate if both equations are in Ax + By = C form.
What should you do if your answer comes out as an ugly non-integer?
Suspect a copying error — recheck the numbers, and always verify by plugging back into both equations.
