Question

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

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

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

Explanation:

Step1: Simplify the second equation

Divide the second equation \(6x + 6y = 18\) by 6, we get \(x + y = 3\), which can be rewritten as \(y = 3 - x\).

Step2: Substitute \(y\) into the first equation

Substitute \(y = 3 - x\) into \(-7x - 2y = 14\), we have \(-7x - 2(3 - x) = 14\).
Expand the left side: \(-7x - 6 + 2x = 14\).
Combine like terms: \(-5x - 6 = 14\).
Add 6 to both sides: \(-5x = 14 + 6 = 20\).
Divide both sides by -5: \(x = \frac{20}{-5} = -4\).

Step3: Find the value of \(y\)

Substitute \(x = -4\) into \(y = 3 - x\), we get \(y = 3 - (-4) = 3 + 4 = 7\)? Wait, but the graph shows \(y = 4\). Wait, maybe I made a mistake. Wait, let's check the graph again. The graph has a horizontal line at \(y = 4\). Let's use the first equation to find \(x\) when \(y = 4\).
Substitute \(y = 4\) into \(-7x - 2y = 14\), we have \(-7x - 2\times4 = 14\), so \(-7x - 8 = 14\), \(-7x = 14 + 8 = 22\), \(x = -\frac{22}{7}\)? No, that's not right. Wait, maybe the second equation: when \(y = 4\), \(6x + 6\times4 = 18\), \(6x + 24 = 18\), \(6x = -6\), \(x = -1\). Wait, the graph has two green dots: one at \(x = -1\), \(y = 4\) and one at \(x = 0\), \(y = 4\)? Wait, no, the graph's x-axis: let's see the x-axis marks. Wait, 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: \(6x + 6y = 18\), \(y = -x + 3\). Now, set them equal: \(-\frac{7}{2}x - 7 = -x + 3\). Multiply both sides by 2: \(-7x - 14 = -2x + 6\). Bring variables to left and constants to right: \(-7x + 2x = 6 + 14\), \(-5x = 20\), \(x = -4\). Then \(y = -(-4) + 3 = 7\). But the graph shows \(y = 4\). Wait, maybe the graph is not for these equations? Wait, the problem says "you can use the interactive graph below to find the solution". Maybe the graph has the intersection at \(x = -4\), \(y = 7\)? No, the graph's y-axis is up to 7? Wait, the graph's y-axis has marks 1,2,3,4,5,6,7. Wait, the green dots: one at \(x = -1\), \(y = 4\) and one at \(x = 0\), \(y = 4\)? No, maybe the graph is a horizontal line \(y = 4\) and another line. Wait, maybe I made a mistake in solving. Let's try again.

First equation: \(-7x - 2y = 14\)

Second equation: \(6x + 6y = 18\) => \(x + y = 3\) => \(y = 3 - x\)

Substitute into first equation:

\(-7x - 2(3 - x) = 14\)

\(-7x - 6 + 2x = 14\)

\(-5x - 6 = 14\)

\(-5x = 20\)

\(x = -4\)

Then \(y = 3 - (-4) = 7\)

But the graph shows \(y = 4\). Maybe the graph is incorrect, or I misread the equations. Wait, the first equation: is it \(-7x - 2y = 14\) or \(-7x + 2y = 14\)? If it's \(-7x + 2y = 14\), then \(2y = 7x + 14\), \(y = \frac{7}{2}x + 7\). Then with \(y = 4\), \(4 = \frac{7}{2}x + 7\), \(\frac{7}{2}x = -3\), \(x = -\frac{6}{7}\), no. Wait, maybe the second equation is \(6x - 6y = 18\)? Then \(x - y = 3\), \(y = x - 3\). Substitute into first equation: \(-7x - 2(x - 3) = 14\), \(-7x - 2x + 6 = 14\), \(-9x = 8\), \(x = -\frac{8}{9}\), no. Wait, the original problem's graph: the x-axis has marks, maybe the green dots are at \(x = -4\), \(y = 7\) and \(x = 1\), \(y = 4\)? No, I think I should follow the algebraic solution. So \(x = -4\), \(y = 7\). Wait, but the graph shows \(y = 4\). Maybe the graph is for a different system. But according to the equations, the solution is \(x = -4\), \(y = 7\). Wait, no, let's check the second equation again. \(6x + 6y = 18\) => \(x + y = 3\). If \(x = -4\), then \(y = 7\), and \(6\times(-4) + 6\times7 = -24 + 42 = 18\), which is correct. And \(-7\times(-4) - 2\times7 = 28 - 14 = 14\), which is correct. So the solution is \(x = -4\), \(y = 7\). But the graph shows \(y = 4\), maybe the graph is not for this system. So the algebraic solution is correct.

Answer:

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