Question

find the solution to the system of equations. you can use the interactive graph below to find the solution. \

$$\begin{cases} -7x - 2y = 14 \\\\ 6x + 6y = 18 \\end{cases}$$

x = \boxed{} y = \boxed{}

Explanation:

Step1: Simplify the second equation

Divide the second equation \(6x + 6y = 18\) by 6: \(x + y = 3\), so \(y = 3 - x\).

Step2: Substitute into the first equation

Substitute \(y = 3 - x\) into \(-7x - 2y = 14\):
\(-7x - 2(3 - x) = 14\)
\(-7x - 6 + 2x = 14\)
\(-5x - 6 = 14\)
\(-5x = 20\)
\(x = -4\).

Step3: Find y

Substitute \(x = -4\) into \(y = 3 - x\):
\(y = 3 - (-4) = 7\)? Wait, no, wait. Wait, let's check again. Wait, maybe I made a mistake. Wait, the graph: the first line (blue) is horizontal at y=4? Wait, no, the user's graph: the blue line is at y=4? Wait, no, maybe my substitution is wrong. Wait, let's re - do the equations.

Wait, original equations:

1. \(-7x - 2y = 14\)
2. \(6x + 6y = 18\) → divide by 6: \(x + y = 3\) → \(y = 3 - x\).

Substitute into first equation:
\(-7x - 2(3 - x)=14\)
\(-7x - 6 + 2x = 14\)
\(-5x = 14 + 6 = 20\)
\(x = - 4\). Then \(y = 3 - (-4)=7\)? But the graph has a blue line at y=4? Wait, maybe the graph is for a different system? Wait, no, maybe I misread the equations. Wait, the first equation: \(-7x - 2y = 14\). Let's solve for y: \(-2y = 7x + 14\) → \(y=-\frac{7}{2}x - 7\). The second equation: \(y=-x + 3\). Let's find intersection:

Set \(-\frac{7}{2}x - 7=-x + 3\)
\(-\frac{7}{2}x+x=3 + 7\)
\(-\frac{5}{2}x = 10\)
\(x=-4\). Then \(y=-(-4)+3 = 7\)? But the graph shows a blue line at y=4? Wait, maybe the equations are different? Wait, maybe the first equation is \(-7x - 2y = 14\), let's plug x=-4, y=4: \(-7(-4)-24=28 - 8 = 20
eq14\). Oh! I see my mistake. Wait, maybe the second equation is \(6x + 6y = 18\) → x + y = 3, but maybe the first equation was misread. Wait, the user's graph: the blue line is horizontal at y=4, and the purple at y=-4? Wait, no, the green dots: one on blue line (y=4) at x=-1? No, maybe I made a mistake in the problem. Wait, let's start over.

Wait, let's use elimination. Multiply the second equation (x + y = 3) by 2: \(2x + 2y = 6\). Now add to the first equation: \(-7x - 2y+2x + 2y=14 + 6\) → \(-5x = 20\) → \(x=-4\). Then from x + y = 3, y=3 - (-4)=7. But the graph: the blue line is at y=4? That's a contradiction. Wait, maybe the equations are \(-7x - 2y = 14\) and \(6x + 6y = 18\) is wrong? Wait, maybe the second equation is \(6x + 6y = - 18\)? No, the user wrote 18. Wait, maybe the graph is not for this system? Or maybe I misread the first equation. Wait, the first equation: \(-7x - 2y = 14\). Let's solve for y: \(y=-\frac{7}{2}x - 7\). The second equation: \(y=-x + 3\). Let's plot x=-4: y=-\frac{7}{2}(-4)-7=14 - 7 = 7. y=-(-4)+3=7. So the intersection is at (-4,7). But the graph shows a blue line at y=4. Maybe the graph is a mistake, or I misread the equations. Wait, maybe the first equation is \(-7x - 2y = 14\) → when x=0, y=-7. The second equation: x + y = 3 → when x=0, y=3. So the lines should intersect at (-4,7). So maybe the graph in the problem is a different one, but according to the equations, x=-4, y=7? Wait, no, wait, let's check with x=-4, y=4: \(-7(-4)-2*4=28 - 8 = 20
eq14\). x=-4, y=7: \(-7(-4)-27=28 - 14 = 14\), which works. And for the second equation: \(6(-4)+67=-24 + 42 = 18\), which works. So the solution is x=-4, y=7.

Answer:

\(x = - 4\)
\(y = 7\)