Question

-10x - 6y = 6\
-5x - 4y = -6\
(\box, \box)\
submit

Explanation:

Step1: Analyze the relationship between the two equations

We have the first equation \(-10x - 6y = 6\) and the second equation \(-5x - 4y=-6\). Notice that if we factor out a \(2\) from the left - hand side of the first equation, we get \(2(-5x - 3y)=6\). But maybe a better approach is to see how to transform the first equation to get the second or vice - versa. Let's try to find a multiplier for the second equation. If we multiply the second equation \(-5x - 4y=-6\) by \(2\), we get \(2\times(-5x - 4y)=2\times(-6)\), which simplifies to \(-10x-8y = - 12\). But our first equation is \(-10x - 6y = 6\). Wait, maybe the problem is about solving the system of linear equations? Let's assume we want to solve the system of equations:
\[

$$\begin{cases} -10x - 6y=6\\ -5x - 4y=-6 \end{cases}$$

\]
Let's use the elimination method. Multiply the second equation by \(2\) to make the coefficients of \(x\) the same (in terms of magnitude).

Step2: Multiply the second equation by 2

Multiply the equation \(-5x - 4y=-6\) by \(2\). We know that for an equation \(ax + by=c\), multiplying both sides by a non - zero constant \(k\) gives \(kax + kby = kc\). So, \(2\times(-5x)+2\times(-4y)=2\times(-6)\), which simplifies to \(-10x-8y=-12\). Now our system is:
\[

$$\begin{cases} -10x - 6y = 6\\ -10x-8y=-12 \end{cases}$$

\]

Step3: Subtract the two equations

Subtract the second equation from the first equation. \((-10x - 6y)-(-10x - 8y)=6-(-12)\).
Simplify the left - hand side: \(-10x - 6y + 10x + 8y=6 + 12\).
The \(-10x\) and \(+10x\) cancel out, and we have \(2y = 18\).

Step4: Solve for y

Divide both sides of \(2y = 18\) by \(2\). \(\frac{2y}{2}=\frac{18}{2}\), so \(y = 9\).

Step5: Substitute y into one of the original equations

Substitute \(y = 9\) into the second original equation \(-5x-4y=-6\). We get \(-5x-4\times9=-6\).
Simplify: \(-5x - 36=-6\).

Step6: Solve for x

Add \(36\) to both sides: \(-5x=-6 + 36\), so \(-5x = 30\).
Divide both sides by \(-5\): \(\frac{-5x}{-5}=\frac{30}{-5}\), so \(x=-6\).

Answer:

The solution to the system of equations \(

$$\begin{cases}-10x - 6y = 6\\-5x - 4y=-6\end{cases}$$

\) is \(x=-6\) and \(y = 9\). If the problem was about factoring or finding a multiplier, if we consider the first equation \(-10x - 6y = 6\) and the second equation \(-5x - 4y=-6\), and we want to see the relationship, we can say that the first equation can be written as \(2\times(-5x - 3y)=6\) and the second is \(-5x - 4y=-6\). But if we assume the problem is to solve the system, the solution is \(x=-6,y = 9\).