Substitution Method

Sommaire

In short: Substitution method solves a system by isolating one variable in one equation and replacing that variable in the other equation, which leaves a single equation with one unknown.

8th GradeAlgebra8.EE.C.8b

What is Substitution Method?

The substitution method works like swapping a nickname for a real name. If one equation says $a = c + 4$, then everywhere $a$ appears in the other equation you may write $c + 4$ instead, and one of the unknowns disappears.

That leaves a one-variable equation that you already know how to solve. Afterward you put the answer back into the first expression to recover the other variable. The approach is most convenient when a variable is already alone, or has a coefficient of 1.

How to spot it

The substitution method shines when one equation already reads y equals something, or x equals something.
A coefficient of 1 on a variable makes it easy to isolate that variable.
A story gives a direct relationship, such as one price being four dollars more than another.
Adding the equations would not remove a variable, so another strategy is needed.

The method

To start the substitution method, pick an equation and solve it for one variable, if that is not done already.
Write that expression in place of the variable in the other equation, using parentheses.
Solve the resulting single-variable equation.
Put the value into the isolated expression to find the other variable, then check in both equations.

Replace a with c + 4, solve for c, then find a.

Substitution Method: solved example

Problem

At a Seattle theater, an adult ticket $a$ costs $\$4$ more than a child ticket $c$, so $a = c + 4$. A family pays $\$58$ for 2 adult and 3 child tickets: $2a + 3c = 58$. Find both prices.

Solution

Substitute $c + 4$ for $a$: $2(c + 4) + 3c = 58$. Distribute: $2c + 8 + 3c = 58$, so $5c = 50$ and $c = 10$.

Then $a = 10 + 4 = 14$. Check: $2(14) + 3(10) = 58$. A child ticket is $\$10$ and an adult ticket is $\$14$.

Try it

Question 1

Solve $x = y + 4$ and $3x – 2y = 17$.

Show the answer

$3(y + 4) – 2y = 17$ gives $y + 12 = 17$, so $y = 5$ and $x = 9$.

Question 2

Solve $2x + y = 11$ and $y = x – 1$.

Show the answer

$2x + (x – 1) = 11$ gives $3x = 12$, so $x = 4$ and $y = 3$.

Substitution Method vs. Elimination Method

Substitution swaps an expression for a variable, so it likes equations where a variable is isolated. Elimination adds or subtracts whole equations to cancel a variable, so it likes equations lined up in the form Ax + By = C. Both reach the same answer; they differ in which form makes the first step easy.

Elimination Method

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