Definition: The substitution method solves a system of equations by solving one equation for a variable and substituting that expression into the other equation.
When does substitution work best?
It is the best choice when one equation already has a variable alone, such as y = 3x – 4, or when a variable has coefficient 1 or -1.
What are the steps?
Solve one equation for one variable. Substitute that expression into the other equation, replacing the variable. Solve the resulting one-variable equation. Back-substitute to find the other variable. Check in both equations.
Example. Solve y = 2x – 7 and 5x + 3y = 20. Replace y: 5x + 3(2x – 7) = 20, so 5x + 6x – 21 = 20 and 11x = 41, giving x = 41/11. Then y = 82/11 – 77/11 = 5/11. The solution is (41/11, 5/11).
What can go wrong?
Forgetting parentheses around the substituted expression, so the 3 multiplies only 2x. Another mistake is substituting back into the equation just solved for y, which proves nothing. If the variables vanish and you get a false statement like 3 = 8, the lines are parallel and there is no solution; a true statement like 4 = 4 means infinitely many.