Systems of Linear Equations • Topic 4 of 4

Word Problems

Many SAT word problems hide a system of two equations. The method is to name two variables, write one equation for each piece of information (often a total and a relationship), then solve by substitution or elimination. Typical setups are "two items with a total cost and total count" or "sum and difference" of two numbers. Define your variables clearly and label units so the equations match the words. After solving, check both conditions. Reliable translation from sentences to two equations is what these high-value SAT questions reward.

✅ Solved examples

1. Two numbers add to 20 and differ by 6. Find them.
x + y = 20, x − y = 6; add: 2x = 26, x = 13, y = 7.
2. Pens cost $2 and notebooks $3. 5 items cost $13. How many of each?
p + n = 5, 2p + 3n = 13. From the first, p = 5 − n; substitute: 10 − 2n + 3n = 13, n = 3, p = 2.
3. The sum of two numbers is 30; one is twice the other. Find them.
x + y = 30, x = 2y; substitute: 3y = 30, y = 10, x = 20.
4. Tickets cost $8 for adults and $5 for children. 4 tickets cost $26 in total. How many of each?
Let a + c = 4 and 8a + 5c = 26. Substitute a = 4 − c: 8(4 − c) + 5c = 26 → 32 − 3c = 26 → c = 2, so a = 2. Two adult and two child tickets.

✏️ Practice — try these, take hints as needed

1. Two numbers add to 18 and differ by 4. Find them.
x + y = 18, x − y = 4.
Add: 2x = 22.
Find x then y.
11 and 7.
2. The sum of two numbers is 24; one is three times the other.
x + y = 24, x = 3y.
Substitute: 4y = 24.
Find y then x.
18 and 6.
3. Apples $2, bananas $1. 6 fruits cost $9. How many of each?
a + b = 6, 2a + b = 9.
Subtract: a = 3.
Find b.
3 apples, 3 bananas.
4. Two numbers add to 50; their difference is 10. Find the larger.
Larger = (sum + difference)/2.
(50 + 10)/2.
Compute.
30.
5. A class has 30 students with twice as many girls as boys. How many boys?
b + g = 30, g = 2b.
b + 2b = 30.
Solve.
10 boys.

📝 Topic test — 8 questions

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