https://file.aiquickdraw.com/mathbot/file/1725403462085avd4fq8p.jpg

The image contains the following system of linear equations:

\[
\begin{aligned}
-2x - 2y &= 30 \quad \text{(Equation 1)} \\
5x + 3y &= 7 \quad \text{(Equation 2)}
\end{aligned}
\]

To solve this system, we can use either the substitution method, elimination method, or matrix method. Let's proceed with the elimination method.

### Step 1: Simplify Equation 1
Equation 1 can be simplified by dividing everything by -2:

\[
\begin{aligned}
x + y &= -15 \quad \text{(Equation 1 Simplified)}
\end{aligned}
\]

### Step 2: Express \( y \) in terms of \( x \) from the simplified Equation 1
\[
\begin{aligned}
y &= -15 - x \quad \text{(Equation 3)}
\end{aligned}
\]

### Step 3: Substitute Equation 3 into Equation 2
\[
\begin{aligned}
5x + 3(-15 - x) &= 7 \\
5x - 45 - 3x &= 7 \\
2x - 45 &= 7 \\
2x &= 52 \\
x &= 26
\end{aligned}
\]

### Step 4: Substitute \( x = 26 \) back into Equation 3 to find \( y \)
\[
\begin{aligned}
y &= -15 - 26 \\
y &= -41
\end{aligned}
\]

### Solution:
The solution to the system of equations is \( x = 26 \) and \( y = -41 \).

Would you like more details or have any questions? Here are some related questions:

1. What are the other methods to solve this system of equations?
2. How can you check if the solution is correct?
3. What happens if we graph these two equations?
4. What are the conditions for a system of linear equations to have no solution?
5. How do you solve a system of linear equations using matrices?

**Tip:** Always double-check your solutions by substituting them back into the original equations.

Learn how to solve the system of linear equations -2x - 2y = 30 and 5x + 3y = 7 using the elimination method. This detailed step-by-step guide covers key algebraic concepts and methods suitable for students in Grades 8-10.

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