https://file.aiquickdraw.com/mathbot/file/1726665685166kzb9rzqw.jpg

The system of equations given in the image is:

\[
\begin{cases}
-4x + y = 6 \\
-5x - y = 21
\end{cases}
\]

We can solve this system using either substitution or elimination. I'll use the elimination method for clarity:

### Step 1: Add the two equations
First, we add the two equations to eliminate \( y \).

\[
(-4x + y) + (-5x - y) = 6 + 21
\]

This simplifies to:

\[
-9x = 27
\]

### Step 2: Solve for \( x \)
Solve for \( x \) by dividing both sides by \(-9\):

\[
x = \frac{27}{-9} = -3
\]

### Step 3: Substitute \( x = -3 \) back into one of the original equations
Substitute \( x = -3 \) into the first equation, \( -4x + y = 6 \):

\[
-4(-3) + y = 6
\]

Simplifying:

\[
12 + y = 6 \quad \Rightarrow \quad y = 6 - 12 = -6
\]

### Solution
The solution to the system is:

\[
x = -3, \quad y = -6
\]

Would you like further details or explanations on any of the steps?

Here are five related questions to explore further:
1. What other methods can be used to solve systems of equations?
2. How do you solve a system of equations when both variables have the same coefficients?
3. Can systems of equations have no solutions? How can you identify such cases?
4. How does the substitution method differ from the elimination method?
5. What are real-world applications of solving systems of equations?

**Tip:** When using the elimination method, always check if adding or subtracting the equations will directly eliminate one of the variables to simplify your work.

Solve the system of equations: -4x + y = 6 and -5x - y = 21

Learn how to solve a system of linear equations using the elimination method. This example covers the equations -4x + y = 6 and -5x - y = 21, demonstrating the step-by-step process to find the values of x and y.

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