Solving Systems of Linear Equations using Substitution

To check a solution for a system of equations without graphing, we need to substitute the solution into each equation. Then, we have to determine whether all of the equations in the system are true. Let's consider two examples.

Example $1$

Suppose we are given a system and told the solution is $(1,2).$ $$\{\begin{array}{l}x+y=3\\ 3x-y=1\end{array}$$ Let's check the given solution by substituting $1$ and $2$ for $x$ and $y,$ respectively. Since both equations are true, $(1,2)$ is a solution to the system.

Example $2$

Consider a second example now. $$\{\begin{array}{l}2x+y=1\\ x-2y=3\end{array}$$ Suppose we are told $(1,0)$ is a solution to the system. Let's see if that's correct! Since neither equation is true, $(1,0)$ is not a correct solution to the system. Note that if only one of the equations produces a false statement, the solution is incorrect.