https://file.aiquickdraw.com/mathbot/file/1725952458748u80y5bop.jpg

The system of equations given in the image is:

1. \( 3x - 4y = -5 \)
2. \( x + 6y = -31 \)

Let's solve this system of equations using the substitution or elimination method.

### Step 1: Solve for \( x \) from the second equation
From the second equation:

\[
x + 6y = -31
\]

Solve for \( x \):

\[
x = -31 - 6y
\]

### Step 2: Substitute \( x \) in the first equation
Substitute \( x = -31 - 6y \) into the first equation:

\[
3(-31 - 6y) - 4y = -5
\]

### Step 3: Simplify and solve for \( y \)
Simplifying the equation:

\[
-93 - 18y - 4y = -5
\]

Combine like terms:

\[
-93 - 22y = -5
\]

Add 93 to both sides:

\[
-22y = 88
\]

Divide by -22:

\[
y = -4
\]

### Step 4: Find \( x \)
Substitute \( y = -4 \) back into the expression for \( x \):

\[
x = -31 - 6(-4)
\]

Simplify:

\[
x = -31 + 24 = -7
\]

### Final Answer:
The solution is \( x = -7 \) and \( y = -4 \).

---

Would you like further details or have any questions? Here are five related questions to explore further:

1. What is the graphical interpretation of solving this system of equations?
2. How would you solve this system using the elimination method?
3. Can you verify the solution by substituting the values back into both equations?
4. What are some real-world applications of solving linear systems of equations?
5. How would the solution change if the coefficients of the equations were different?

**Tip:** Always check your solutions by substituting them back into the original equations to ensure accuracy.

Find the value of the unknown variables in the following system of equations: 3x - 4y = -5 and x + 6y = -31.

Learn how to solve the system of linear equations 3x - 4y = -5 and x + 6y = -31 using the substitution method. Follow a detailed step-by-step solution that applies key algebraic concepts. Ideal for students in Grades 9-11.

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