Substitution Method

Solve linear systems using substitution method. Math 10-C Alberta mathematics.

Lesson 5.2 of Systems of Equations in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• The Method. Learn the substitution method for solving systems of equations. Understand the big idea behind substitution, follow the four clear steps from isolating a variable to writing the final ordered pair, and pick up a tip for choosing which variable to isolate first.
• Worked Examples. Work through five detailed examples covering every common scenario: a variable already almost isolated, a one-variable equation, decimal coefficients, fraction coefficients, and fractions built into the original equations. Each example shows every algebraic step with tips for handling fractions and sign errors.

Inside the lesson — a free preview

What is Substitution?

Definition — Substitution Method. The substitution method is an algebraic method of solving a system of equations by isolating one variable and substituting its expression into the other equation.

The big idea: if you can rewrite one equation as y = something (or x = something), you can plug that expression into the other equation. That gives you one equation with only one variable — which you already know how to solve.

The 4 Steps

Before You Start. Number your equations ① and ② before you begin so you can track which equation you are working with at each step.

The 4 Steps of Substitution. Step 1 — Isolate one variable.

Solve for x or y in one of the equations.

Step 2 — Substitute and solve.

Plug that expression into the other equation. Now you have one equation with one variable — solve it.

Step 3 — Back-substitute.

Take the value you just found and substitute it back in to find the other variable.

Step 4 — Write the ordered pair.

State the answer as an ordered pair (x, y) that satisfies both original equations.

Tip — Which Variable Should You Isolate?. Before starting, look at both equations.

Pick the variable that has a coefficient of 1 or -1 — isolating it will not introduce any fractions and keeps the arithmetic clean.

If no variable has a coefficient of 1 or -1, you may end up working with fractions (which is fine — just be careful).

Worked example: A Variable Already Almost Isolated

Example 1 — Solve the System. ① 4x + 5y = 26

② 3x = y - 9

Solution. Step 1 — Isolate a variable.

Equation ② is almost solved for y already. Add 9 to both sides:

3x + 9 = y

So y = 3x + 9.

Step 2 — Substitute (3x + 9) for y in equation ①.

4x + 5(3x + 9) = 26

4x + 15x + 45 = 26

19x + 45 = 26

19x = 26 - 45

19x = -19

x = -1

Step 3 — Plug x = -1 back into y = 3x + 9.

y = 3(-1) + 9

y = -3 + 9

y = 6

Step 4 — Write the ordered pair.

Solution: (-1, 6)

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