Question

solve the system by graphing. \

$$\begin{cases} 6x + 3y = -9 \\\\ -2x + y = 1 \\end{cases}$$

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

Explanation:

Step1: Rewrite equations in slope - intercept form

For the first equation \(6x + 3y=-9\), solve for \(y\):
Subtract \(6x\) from both sides: \(3y=-6x - 9\).
Divide by 3: \(y=-2x - 3\). The slope \(m_1=-2\) and the \(y\) - intercept \(b_1 = - 3\).

For the second equation \(-2x + y = 1\), solve for \(y\):
Add \(2x\) to both sides: \(y = 2x+1\). The slope \(m_2 = 2\) and the \(y\) - intercept \(b_2=1\).

Step2: Find intersection point (solution)

We can also solve the system by substitution or elimination. Let's use elimination.
The system is:
\(

$$\begin{cases}6x + 3y=-9\\-2x + y = 1\end{cases}$$

\)
Multiply the second equation by 3: \(-6x+3y = 3\)
Now we have the two equations:
\(

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

\)
Add the two equations together:
\((6x-6x)+(3y + 3y)=-9 + 3\)
\(6y=-6\)
Divide both sides by 6: \(y=-1\)
Substitute \(y = - 1\) into the second original equation \(-2x+y = 1\):
\(-2x-1 = 1\)
Add 1 to both sides: \(-2x=2\)
Divide by \(-2\): \(x=-1\)

Answer:

The solution to the system is \(x=-1,y = - 1\) or the ordered pair \((-1,-1)\)