GED Mathematical Reasoning
Systems of Equations
A system is two equations with the same two unknowns. One equation alone has too many possible answers; two together pin down exactly one pair. On the test these are almost always two-item word problems — adult and child tickets, coffee and muffins.
The words first
- system
- two or more equations that must both be true at once
- substitution
- solve one equation for one letter, then plug it into the other
- elimination
- add or subtract the equations to make one letter disappear
The method
ELIMINATION — usually fastest on the test
x + y = 10 ← total number of things
3x + 5y = 38 ← total cost
1. Make one letter match in size.
Multiply the top equation by 3: 3x + 3y = 30
2. Subtract one equation from the other so that letter cancels.
3x + 5y = 38
− 3x + 3y = 30
————————————————
2y = 8
3. Solve what is left. y = 4
4. Put it back in the easy equation.
x + 4 = 10 → x = 6
SUBSTITUTION — good when one equation is already solved
y = 2x + 1 → put (2x + 1) wherever y appears in the other one.
Worked all the way through
The problem
Adult tickets cost $9 and child tickets cost $5. Twelve tickets were sold for $80 total. How many adult tickets?
- Step 1 — Name both unknowns.a = adult tickets, c = child tickets.
- Step 2 — Write the count equation.Twelve tickets altogether: a + c = 12
- Step 3 — Write the money equation.Each adult brings $9, each child $5, totalling $80: 9a + 5c = 80
- Step 4 — Make the c terms match. Multiply the first equation by 5.5a + 5c = 60
- Step 5 — Subtract that from the money equation so c cancels.(9a + 5c) − (5a + 5c) = 80 − 60 → 4a = 20
- Step 6 — Solve.a = 5
- Step 7 — Check both facts.5 adults and 7 children is 12 tickets ✓, and 9(5) + 5(7) = 45 + 35 = 80 ✓
On the board
a + c = 12
12 tickets altogether
9a + 5c = 80
$80 altogether
5a + 5c = 60
first equation × 5, to match the 5c
(9a + 5c) − (5a + 5c) = 80 − 60
subtract, and c cancels
4a = 20
a = 5
divide both sides by 4
Answer
5 adult tickets
A second one, in a different form
If x + y = 18 and x − y = 4, what is x?
x + y = 18
x − y = 4
2x = 22
ADD the equations — the y's cancel
x = 11
divide both sides by 2
11 + y = 18 → y = 7
put it back to find y
Answer
x = 11
The usual mistake
Solving for one variable and forgetting the question asked for the other.
Check yourself
Always check your answer against BOTH original sentences. An answer that fits the count but not the money means an arithmetic slip somewhere.
Practice it
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