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 $\{\begin{array}{l}3x+2y=16\\ 7x+y=19\end{array}$
Solve the second equation for $y$ .
$y=19-7x$
Substitute $19-7x$ for $y$ in the first equation and solve for $x$ .
$\begin{array}{l}3x+2(19-7x)=16\\ 3x+38-14x=16\\ -11x=-22\\ x=2\end{array}$
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 $(2,5)$ .
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, $0=0$ .