EDGE Math 8

Substitution Method

To solve a system of two linear equations in two variables using the substitution method, solve for one variable in either of the given equations. Then substitute the obtained expression for that variable in the other equation.

Example 1

Solve the system of linear equations using the substitution method.

{\begin{array}{l} x + 3 y = 11 \\ 2 x + y = 7 \end{array}

Solution

The coefficient of the variable x in the first equation is 1; thus, it is easier to solve for x in terms of y using the first equation.

Using the first equation, solve for x in terms of y.

x = 11 - 3 y \to equation (a)

Substitute $11 - 3 y$ for x in the second equation and solve for y.

$2 (11 - 3 y) + y$	$= 7$
$22 - 6 y + y$	$= 7$
$- 5 y + 22$	$= 7$
$- 5 y$	$= 7 - 22$
$- 5 y$	$= - 15$
$y$	$= \frac{- 15}{- 5}$
$y$	$= 3$

Thus, $y = 3.$ To solve for x, substitute 3 for y in equation (a).

x = 11 - 3 (3) = 11 - 9 = 2

Therefore, $x = 2$ and $y = 3.$

Checking

Substitute 2 for x and 3 for y in the given equations of the system.

$x + 3 y$	$= 11$
$2 + 3 (3)$	$\overset{?}{=} 11$
$2 + 9$	$\overset{?}{=} 11$
$11$	$= 11 True$

$2 x + y$	$= 7$
$2 (2) + 3$	$\overset{?}{=} 7$
$4 + 3$	$\overset{?}{=} 7$
$7$	$= 7 True$

Since both equations are satisfied, the solution is $(2, 3) .$

Always remember that you also need to solve for the value of the other variable. You should give not just one number, but an ordered pair.