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 = \square\\) \\(y = \square\\)

Explanation:

Step1: Simplify the second equation

Divide the second equation \(6x + 6y = 18\) by 6:
\(\frac{6x}{6}+\frac{6y}{6}=\frac{18}{6}\)
\(x + y = 3\)
We can express \(y\) as \(y = 3 - x\).

Step2: Substitute \(y = 3 - x\) into the first equation

The first equation is \(-7x - 2y = 14\). Substitute \(y\):
\(-7x - 2(3 - x) = 14\)

Step3: Expand and solve for \(x\)

Expand the left - hand side:
\(-7x-6 + 2x = 14\)
Combine like terms:
\(-5x-6 = 14\)
Add 6 to both sides:
\(-5x=14 + 6\)
\(-5x = 20\)
Divide both sides by \(-5\):
\(x=\frac{20}{-5}=-4\)

Step4: Find \(y\) using \(y = 3 - x\)

Substitute \(x=-4\) into \(y = 3 - x\):
\(y=3-(-4)=3 + 4 = 7\)? Wait, no, wait. Wait, maybe I made a mistake. Wait, let's check the graph. Wait, the blue line and pink line. Wait, maybe my substitution is wrong. Wait, let's re - do the second equation. Wait, \(6x + 6y = 18\), divide by 6: \(x + y=3\), correct. First equation: \(-7x-2y = 14\). Let's solve the system by elimination. Multiply the second equation by 2: \(2x + 2y=6\). Now add to the first equation: \((-7x-2y)+(2x + 2y)=14 + 6\). So \(-5x=20\), so \(x=-4\). Then from \(x + y=3\), \(y=3 - (-4)=7\)? But the graph has horizontal lines. Wait, maybe the equations are different. Wait, maybe 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 find intersection: \(-\frac{7}{2}x-7=-x + 3\). Multiply both sides by 2: \(-7x-14=-2x + 6\). \(-7x + 2x=6 + 14\), \(-5x=20\), \(x=-4\). Then \(y=-(-4)+3=7\)? But the graph shows horizontal lines. Wait, maybe the original equations are misread. Wait, the user's graph: blue line and pink line. Wait, maybe the first equation is \(-7x-2y = 14\), let's solve for \(y\): \(2y=-7x - 14\), \(y=-\frac{7}{2}x-7\). The second equation: \(6x + 6y = 18\), \(y=-x + 3\). The intersection is at \(x=-4\), \(y = 7\)? But the graph has points at \(x=-1,y = 4\) and \(x = 1,y = 4\) (blue), and \(x=-1,y=-4\), \(x = 1,y=-4\) (pink). Wait, maybe I misread the equations. Wait, maybe the first equation is \(-7x-2y = 14\) is wrong. Wait, maybe it's \(-7x+2y = 14\)? No, the user wrote \(-7x - 2y = 14\). Wait, let's check with \(x=-4\), \(y = 7\): plug into first equation: \(-7(-4)-27=28 - 14 = 14\), correct. Second equation: \(6(-4)+67=-24 + 42 = 18\), correct. So the solution is \(x=-4\), \(y = 7\)? But the graph's horizontal lines: maybe the graph is just a tool, and the calculation is correct.

Answer:

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