Question

solve the system by graphing.\

$$\begin{cases}-4x + 3y = 24\\\\x + 3y = 9\\end{cases}$$

\dots\
use the graphing tool to graph the system.\
click to enlarge graph

Explanation:

Step1: Rewrite equations in slope-intercept form

For \(-4x + 3y = 24\):
Add \(4x\) to both sides: \(3y = 4x + 24\)
Divide by 3: \(y=\frac{4}{3}x + 8\)

For \(x + 3y = 9\):
Subtract \(x\) from both sides: \(3y=-x + 9\)
Divide by 3: \(y = -\frac{1}{3}x+3\)

Step2: Find intersection (solve the system algebraically to check)

Subtract the second equation from the first:
\((\frac{4}{3}x + 8)-(-\frac{1}{3}x + 3)=0\)
\(\frac{4}{3}x+\frac{1}{3}x+8 - 3 = 0\)
\(\frac{5}{3}x+5 = 0\)
\(\frac{5}{3}x=-5\)
\(x=-3\)

Substitute \(x = -3\) into \(y = -\frac{1}{3}x + 3\):
\(y=-\frac{1}{3}(-3)+3 = 1 + 3 = 4\)

Answer:

The solution to the system is \(x=-3\), \(y = 4\) (the point of intersection of the two lines is \((-3,4)\)).