XMocks · SAT Math

Solve a system of linear equations by elimination

A system is a pair of equations that describe the same two unknowns. Its solution must make both statements true. This lesson shows how to define the quantities, eliminate one variable, and check the answer against the original situation.

1. Define each quantity before writing equations

A school store sells notebooks and pens. Two notebooks and three pens cost $14. Four notebooks and one pen cost $18. Let n be the price of one notebook in dollars and p the price of one pen in dollars.

2n + 3p = 14
4n + p = 18

The coefficients count items in each purchase. The right side is the total cost in dollars. Keeping those units in view helps catch a swapped quantity or a missing item.

2. Make one variable cancel

The second equation has one p. Multiply every term in that equation by 3 so both equations contain 3p:

3(4n + p = 18)  →  12n + 3p = 54

Subtract the first equation from the new equation. The pen terms cancel: 10n = 40, so n = 4. Substitute 4 for n in 4n + p = 18: 16 + p = 18, so p = 2.

Check both original purchases

  • 2 notebooks at $4 and 3 pens at $2: $8 + $6 = $14.
  • 4 notebooks at $4 and 1 pen at $2: $16 + $2 = $18.

The pair (n, p) = (4, 2) satisfies both equations, so the item prices are $4 per notebook and $2 per pen.

3. Three checks that prevent common mistakes

  1. Scale every term. Multiplying the second equation by 3 changes 4n + p = 18 to 12n + 3p = 54; the total on the right changes too.
  2. Track subtraction signs. Subtracting 2n + 3p from 12n + 3p leaves 10n, because the first n coefficient is 12.
  3. Verify in the original context. Put both values into both equations and check the units. A pair that works in only one equation is not the system's solution.

If the coefficients do not line up, you can multiply one or both equations by a nonzero number to create matching coefficients. Substitution is another option when one equation already isolates a variable. Choose the method that keeps the arithmetic clear.

Try one: compare two ticket orders

Two student tickets and one adult ticket cost $19. Four student tickets and one adult ticket cost $31. Find each ticket price. Define s as the student price and a as the adult price, then write both equations before opening the answer.

Show the worked answer

The equations are 2s + a = 19 and 4s + a = 31. Subtract the first from the second: 2s = 12, so s = 6. Then 12 + a = 19, so a = 7. Check: 2(6) + 7 = 19 and 4(6) + 7 = 31. A student ticket costs $6 and an adult ticket costs $7.

Choose focused practice

The College Board's SAT Math question bank lists systems of two linear equations as a skill. Use its Math filters to find more questions, then review the reasoning for each answer.

This independently prepared lesson is an XMocks study resource and is not affiliated with or endorsed by the College Board. For current SAT format and policies, use College Board's official resources.