Solving Systems of Linear Equations Using Substitution

A system of linear equations is just a set of two or more linear equations.

In two variables $(x and y)$ , the graph of a system of two equations is a pair of lines in the plane.

There are three possibilities:

The lines intersect at zero points. (The lines are parallel.)
The lines intersect at exactly one point. (Most cases.)
The lines intersect at infinitely many points. (The two equations represent the same line.)

How to Solve a System Using The Substitution Method

Step $1$ : First, solve one linear equation for $y$ in terms of $x$ .
Step $2$ : Then substitute that expression for $y$ in the other linear equation. You'll get an equation in $x$ .
Step $3$ : Solve this, and you have the $x$ -coordinate of the intersection.
Step $4$ : Then plug in $x$ to either equation to find the corresponding $y$ -coordinate.

Note $1$ : If it's easier, you can start by solving an equation for $x$ in terms of $y$ , also – same difference!

Example:

Solve the system ${\begin{cases} 3 x + 2 y = 16 \\ 7 x + y = 19 \end{cases}$

Solve the second equation for $y$ .

$y = 19 - 7 x$

Substitute $19 - 7 x$ for $y$ in the first equation and solve for $x$ .

$\begin{array}{l} 3 x + 2 (19 - 7 x) = 16 \\ 3 x + 38 - 14 x = 16 \\ - 11 x = - 22 \\ x = 2 \end{array}$

Substitute $2$ for $x$ in $y = 19 - 7 x$ and solve for $y$ .

$\begin{array}{l} y = 19 - 7 (2) \\ y = 5 \end{array}$

(2, 5)

Note $2$ : If the lines are parallel, your $x$ -terms will cancel in step $2$ , and you will get an impossible equation, something like $0 = 3$ .

Note $3$ : If the two equations represent the same line, everything will cancel in step $2$ , and you will get a redundant equation, $0 = 0$ .