Question

solve the system by graphing.\

$$\begin{cases}2x + 3y = 6\\\\6x = -9y + 18\\end{cases}$$

\use the graphing tool to graph the system.

Explanation:

Step1: Rewrite equations in slope - intercept form

For the first equation \(2x + 3y=6\), solve for \(y\):
Subtract \(2x\) from both sides: \(3y=-2x + 6\)
Divide by 3: \(y=-\frac{2}{3}x + 2\)

For the second equation \(6x=-9y + 18\), solve for \(y\):
Add \(9y\) to both sides: \(9y+6x = 18\)
Subtract \(6x\) from both sides: \(9y=-6x + 18\)
Divide by 9: \(y=-\frac{6}{9}x+2=-\frac{2}{3}x + 2\)

Step2: Analyze the two equations

Since both equations are equivalent (they have the same slope \(-\frac{2}{3}\) and the same y - intercept \(2\)), their graphs are the same line.

Answer:

The system of equations has infinitely many solutions, and the two lines coincide (they are the same line \(y =-\frac{2}{3}x + 2\)).