# Solving Systems of Linear Equations Using Substitution

### Systems of Linear equations:

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

In two variables $\left(x\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y\right)$ , 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 $\left\{\begin{array}{l}3x+2y=16\\ 7x+y=19\end{array}$

Solve the second equation for $y$ .

$y=19-7x$

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

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

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

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

The solution is $\left(2,5\right)$ .

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, $0=0$ .