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Step 10: Pre-algebra

Systems of two linear equations

Step 10 · Pre-algebra topics

Solve a pair of linear equations by graphing, substitution and elimination, know when there is one solution, none or infinitely many, and solve word problems with two unknowns.

Solving by graphing

A system is two equations that must both be true at once. Each equation is a line, and the solution is the point where the lines cross: it lies on both.

y = x + 1 and y = −x + 7 cross at (3, 4), the solution.

Example: Checking a point

Is (3, 4) the solution of y = x + 1 and y = −x + 7?

  1. First equation: 3 + 1 = 4, so it fits.
  2. Second equation: −3 + 7 = 4, so it fits too.

AnswerYes

The linesSolutions
cross onceone
are parallel (same slope, different intercepts)none
are the same lineinfinitely many

Solving by substitution

Substitution: get one letter on its own in one equation, then put that expression in its place in the other equation.

Example: Substitution

Solve y = 2x − 1 and 3x + y = 14.

  1. Put 2x − 1 in place of y: 3x + 2x − 1 = 14.
  2. 5x = 15, so x = 3.
  3. y = 2 × 3 − 1 = 5.

Answer(3, 5)

Example: Getting a letter alone first

Solve x + 2y = 7 and 2x − y = 4.

  1. From the first: x = 7 − 2y.
  2. Substitute: 2(7 − 2y) − y = 4, so 14 − 5y = 4.
  3. y = 2, and x = 7 − 4 = 3.

Answer(3, 2)

Solving by elimination

Elimination: add or subtract the equations so that one letter cancels out.

Example: Adding

Solve x + y = 10 and x − y = 4.

  1. Add the equations: 2x = 14, so x = 7.
  2. Then 7 + y = 10, so y = 3.

Answer(7, 3)

Example: Multiplying first

Solve 2x + 3y = 12 and x + y = 5.

  1. Double the second: 2x + 2y = 10.
  2. Subtract it from the first: y = 2.
  3. Then x + 2 = 5, so x = 3.

Answer(3, 2)

Example: A word problem

Adult tickets cost $8 and student tickets $5. 20 tickets cost $130. How many of each were sold?

  1. a + s = 20 and 8a + 5s = 130.
  2. Multiply the first by 5: 5a + 5s = 100.
  3. Subtract: 3a = 30, so a = 10 and s = 10.

Answer10 adult and 10 student tickets

Try it yourself

Work each one on paper first, then open the answer to check.

  1. Is (2, 5) a solution of y = 3x − 1 and y = x + 3?

    Show answer

    Yes: 3 × 2 − 1 = 5 and 2 + 3 = 5.

  2. Solve y = x + 2 and y = 3x − 4.

    Show answer

    (3, 5)

  3. Solve x + y = 12 and x − y = 2.

    Show answer

    (7, 5)

  4. Solve 3x + 2y = 16 and x + 2y = 8.

    Show answer

    (4, 2)

  5. How many solutions do y = 2x + 1 and y = 2x − 3 have?

    Show answer

    None: the lines are parallel.

  6. Two numbers add to 30 and differ by 8. What are they?

    Show answer

    19 and 11

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