Question

GRAPHING LINEAR SYSTEMS Use the graphing method to solve the linear system and describe its solution(s). $$ \begin{aligned} &4 x+2 y=12\\\ &-6 x+3 y=6 \end{aligned} $$

Short answer

Graphically, the line \(y = -2x + 6\) and \(y = 2x + 2\) intersect at the point (1, 4). Therefore, the solution to the system of equations is the point (1, 4).

Step-by-step solution

Rewrite in Slope-Intercept Form

First, rewrite each equation in the y = mx + b form. For the first equation, \(4x + 2y = 12\), divide every term by 2. This gives \(2x + y = 6\), or \(y = -2x + 6\). For the second equation, \(-6x + 3y = 6\), divide every term by 3. This gives \(-2x + y = 2\), or \(y = 2x + 2\).

Graph the Equations

The next step is to graph these equations. You could do it by choosing values for x and then solving for y. Make sure to choose at least two values for x so you can graph a line. The line for the equation \(y = -2x + 6\) will be downward sloping, and the line for \(y = 2x + 2\) will be upward sloping.

Find the Point of Intersection

The solution to the system of equations is the point where the lines intersect. If the lines cross at a specific point, then that point is the unique solution. If the two lines are parallel (and therefore do not intersect), then there are no solutions. If the two lines coincide, then there are infinitely many solutions.