Question

Use a graph to solve the linear system. Check your solution algebraically. (Review 7.1 ) $$\begin{aligned}&\frac{1}{2} x+3 y=18\\\&2 x+6 y=-12\end{aligned}$$

Short answer

The solution to the system of equations is the point of intersection of the two lines on the graph. These points, when substituted into the original equations should satisfy all equations.

Step-by-step solution

Arrange Equations in y=

Solving each equation for y will give a clearer representation on a graph. For the first equation \(\frac{1}{2} x+3 y - 18 = 0\), the equation can be re-arranged to \(y = \frac{18 - \frac{1}{2} x}{3}\). In a similar process, the second equation \(2 x+6 y + 12 = 0\) can be re-arranged to \(y = \frac{-12 - 2x}{6}\).

Plot the Linear Relations

Plot the rearranged equations on the graph. Values for x will be substituted into each equation, and the corresponding values for y noted.

Find Intersection Point

Observe where the two lines intersect on the graph. The coordinates at this point of intersection would be the solution to the system.

Check Solution Algebraically

Substitute the determined values for x and y into both original equations to verify if they hold true. If they do, then the graphical solution is correct.