Solving systems by substitution

Solving systems by substitution

Solving systems by substitution

Substitution is the best method when one of the equations already has a variable on its own.

The idea

If one equation tells you what y equals, you can replace y in the other equation with that expression. The second equation then has only one variable, which you can solve.

A worked example

y = 2x − 1

3x + y = 9

Replace y in the second equation with 2x − 1:

3x + (2x − 1) = 9 → 5x − 1 = 9 → 5x = 10 → x = 2.

Now put x = 2 back into the first equation: y = 2·2 − 1 = 3. The solution is (2, 3).

Always check

Put your (x, y) into both original equations. If both are true, you are right.

Three rules that always help

  • Use substitution when a variable is already isolated.
  • Replace that variable in the other equation, then solve for the remaining one.
  • Substitute back to find the second value, and check in both equations.

Keep going