Question

solve the following system of equations by graphing. then determine whether the system is consistent or inconsistent and whether the equations are dependent or independent. if the system is consistent, give the solution.
\

$$\begin{cases} -6x + 6y = 12 \\\\ 2x + 2y = -24 \\end{cases}$$

answer

graph the linear equations by writing the equations in slope-intercept form:

$y = \underline{\quad} x + \underline{\quad}$

$y = \underline{\quad} x + \underline{\quad}$

determine whether the system is consistent or inconsistent. if it is consistent, give the solution.

\\(\circ\\) consistent

\\(\circ\\) inconsistent

Explanation:

Step1: Convert first equation to slope - intercept form

We have the equation \(-6x + 6y=12\).
First, add \(6x\) to both sides: \(6y = 6x+12\).
Then divide both sides by \(6\): \(y=\frac{6x + 12}{6}=x + 2\).

Step2: Convert second equation to slope - intercept form

We have the equation \(2x+2y=-24\).
First, subtract \(2x\) from both sides: \(2y=-2x - 24\).
Then divide both sides by \(2\): \(y=\frac{-2x-24}{2}=-x - 12\).

Step3: Determine consistency

To find the solution of the system \(

$$\begin{cases}y=x + 2\\y=-x-12\end{cases}$$

\), we set the two equations equal to each other:
\(x + 2=-x-12\)
Add \(x\) to both sides: \(2x+2=-12\)
Subtract \(2\) from both sides: \(2x=-14\)
Divide by \(2\): \(x=-7\)
Substitute \(x = - 7\) into \(y=x + 2\), we get \(y=-7 + 2=-5\).
Since there is a unique solution \((-7,-5)\), the system is consistent.

Answer:

For the first equation \(y = 1x+2\) (slope - intercept form \(y=x + 2\)), for the second equation \(y=-1x-12\) (slope - intercept form \(y=-x - 12\)). The system is Consistent, and the solution is \(x=-7,y = - 5\) (or the ordered pair \((-7,-5)\)).