Skill: Determine whether an ordered pair solves a linear system

An ordered pair $(x,y)$ solves a system when it makes both equations true at the same time. To check, substitute the $x$-coordinate for $x$ and the $y$-coordinate for $y$ in each equation.

Example: Determine whether $(2,-1)$ solves the system

$$\begin{cases} 3x+2y=4\\ x-y=3 \end{cases}$$

Step 1: Check the first equation.

Substitute $x=2$ and $y=-1$:

$$3(2)+2(-1)=4$$ $$6-2=4$$ $$4=4$$

The first equation is true.

Step 2: Check the second equation.

$$2-(-1)=3$$ $$2+1=3$$ $$3=3$$

The second equation is also true.

Because $(2,-1)$ makes both equations true, it is a solution to the system. It represents the point where the two lines intersect.

If an ordered pair makes even one equation false, it is not a solution to the system.