Question: Graph the system below and write its solution. \[ \begin{cases} y = -\frac{1}{3}x + 3 \\ -3x…

Graph the system below and write its solution.

\[ \begin{cases} y = -\frac{1}{3}x + 3 \\ -3x + y = -7 \end{cases} \]

Note that you can also answer “No solution” or “Infinitely many” solutions.

Solution

To solve the system graphically, we must find the intersection of the two lines given by the equations: \[ \begin{cases} y = -\frac{1}{3}x + 3 & \\ -3x + y = -7 & \end{cases} \] First, plot the line given by the equation \(y = -\frac{1}{3}x + 3\). 1. The y-intercept is 3, so a point on the graph is \((0, 3)\). 2. The slope is \(-\frac{1}{3}\), which means for every 3 units moved to the right, the line goes down 1 unit. A second point is \((3, 2)\). Next, plot the second line given by the equation \(-3x + y = -7\). 1. Rewrite it in slope-intercept form: \[ y = 3x - 7 \] 2. The y-intercept is -7, so a point is \((0, -7)\). 3. The slope is \(3\), which means for every 1 unit moved to the right, the line goes up 3 units. A second point is \((1, -4)\). Graph both lines on the coordinate plane. The intersection of these lines represents the solution to the system of equations. By looking at the graph (or solving algebraically), the lines intersect at the point \((3, 2)\). Thus, the solution to the system is \((3, 2)\).