Question

solve the system of equations $-6x - y = 2$ and $-5x - y = 0$ by combining the equations.
1 $(-6x - y = 2)$
-1 $(-5x - y = 0)$
\

$$\begin{align*} -6x - y &= 2 \\\\ 5x + y &= 0 \\\\ \\hline -2 x + 10 y &= \\square \\end{align*}$$

Explanation:

Step1: Combine the x-terms

The x-terms are \(-6x\) and \(5x\). Adding them together: \(-6x + 5x = -x\) (Wait, but the given step has -2x? Wait, no, let's recalculate. Wait, the first equation is \(-6x - y = 2\), the second equation after multiplying by -1 is \(5x + y = 0\). Now, add the two equations:

For the x-coefficients: \(-6x + 5x = -x\)? Wait, no, maybe I misread. Wait, the user's image shows a step where it's \(-6x - y = 2\) and \(5x + y = 0\), and then adding them:

\(-6x + 5x = -x\)? But the image has -2x. Wait, maybe there's a typo, but let's follow the addition of the equations.

Wait, actually, let's do the addition properly. The first equation: \(-6x - y = 2\), second equation (after multiplying by -1): \(5x + y = 0\). Now, add the left sides: \((-6x - y) + (5x + y) = -6x + 5x - y + y = -x + 0 = -x\). Add the right sides: \(2 + 0 = 2\). Wait, but the image has \(-2x + 10y\)? No, that must be a mistake. Wait, maybe the user made a typo in the image, but let's solve the system correctly.

Wait, the system is:

Equation 1: \(-6x - y = 2\)

Equation 2: \(-5x - y = 0\)

To eliminate y, we can subtract equation 2 from equation 1 (or multiply equation 2 by -1 and add). Let's multiply equation 2 by -1: \(5x + y = 0\). Now add to equation 1:

\(-6x - y + 5x + y = 2 + 0\)

Simplify: \(-x = 2\) => \(x = -2\)

Then substitute x = -2 into equation 2: \(-5(-2) - y = 0\) => \(10 - y = 0\) => \(y = 10\)

But the question in the image is about combining the equations to get the sum. Let's do the addition of the two equations after multiplying the second by -1:

First equation: \(-6x - y = 2\)

Second equation (after -1*): \(5x + y = 0\)

Now, add the left sides: \(-6x - y + 5x + y = (-6x + 5x) + (-y + y) = -x + 0 = -x\)

Add the right sides: \(2 + 0 = 2\)

So the combined equation is \(-x = 2\), but the image has a different setup. Wait, maybe the image's "10y" is a mistake, and it's supposed to be 0y (since -y + y = 0). Then the x-term: -6x + 5x = -x, but the image has -2x. Maybe the user intended to multiply the first equation by -1 and the second by 1? Let's check:

If we multiply equation 1 by -1: \(6x + y = -2\)

Equation 2: \(-5x - y = 0\)

Add them: \(6x -5x + y - y = -2 + 0\) => \(x = -2\), which is correct.

But the image shows multiplying the first equation by 1 and the second by -1, leading to \(-6x - y = 2\) and \(5x + y = 0\). Adding these:

\(-6x + 5x = -x\), \(-y + y = 0\), so \(-x = 2\), so \(x = -2\). Then substitute into equation 2: \(-5(-2) - y = 0\) => \(10 - y = 0\) => \(y = 10\).

But the image's " -2 x + 10 y = [ ]" is incorrect. However, if we follow the addition of the two equations as per the image (even with the typo), let's see:

\(-6x - y = 2\)

\(5x + y = 0\)

Adding the x-terms: -6x + 5x = -x (but image has -2x, maybe a mistake)

Adding the y-terms: -y + y = 0 (but image has 10y, mistake)

Adding the constants: 2 + 0 = 2

So the correct combined equation is \(-x = 2\), so x = -2, then y = -5x (from equation 2: -5x - y = 0 => y = -5x). Substitute x = -2: y = -5*(-2) = 10.

But the question in the image is to fill the box, which is the sum of the right sides? Wait, the right sides are 2 and 0, so 2 + 0 = 2. So the box should be 2.

Wait, the image shows:

-6x - y = 2

5x + y = 0

-----------------

-2 x + 10 y = [ ]

But that's incorrect. The correct addition is:

(-6x + 5x) + (-y + y) = 2 + 0

=> -x + 0 = 2

=> -x = 2

But if we ignore the typo in the image and just add the right sides: 2 + 0 = 2. So the box should be 2.

Step2: Solve for x

From the combined equation (after correcting), we have \(-x = 2\), so \(x = -2\)

Step3: Solve for y

Substitute x = -2 into equation 2: \(-5(-2) - y = 0\) => \(10 - y = 0\) => \(y = 10\)

But the question in the image is about the box when combining the equations. The right sides are 2 and 0, so adding them gives 2. So the box should be 2.

Answer:

The value in the box is \(\boxed{2}\) (assuming the combination is adding the right sides, 2 + 0 = 2).