We can solve the given system of equations using either substitution or elimination. The system is:

1. \( 3x - 10y = -46 \)
2. \( 8x + 10y = 24 \)

Let's solve this using the **elimination method** since the coefficients of \( y \) have opposite signs and can be easily eliminated by adding the equations.

### Step 1: Add both equations together

\[
(3x - 10y) + (8x + 10y) = (-46 + 24)
\]

This simplifies to:

\[
3x + 8x = -46 + 24
\]
\[
11x = -22
\]

### Step 2: Solve for \( x \)

\[
x = \frac{-22}{11} = -2
\]

### Step 3: Substitute \( x = -2 \) back into one of the original equations to solve for \( y \). Let's use the first equation:

\[
3x - 10y = -46
\]
Substitute \( x = -2 \):

\[
3(-2) - 10y = -46
\]
\[
-6 - 10y = -46
\]
\[
-10y = -46 + 6
\]
\[
-10y = -40
\]
\[
y = \frac{-40}{-10} = 4
\]

### Final Solution:

\[
x = -2, \quad y = 4
\]

Would you like more details or have any questions?

Here are some related questions:
1. How can we solve this system using substitution?
2. What if the coefficients of \( y \) weren't opposites? How would we handle it?
3. How can you check if the solution is correct by substitution into both equations?
4. What are other methods to solve systems of equations?
5. How would this solution change if the equations were non-linear?

**Tip:** Always check your solution by substituting the values back into both original equations to verify correctness.

Solve the system of equations: 3x − 10y = −46, 8x + 10y = 24

This guide walks through solving the system of linear equations 3x − 10y = −46 and 8x + 10y = 24 using the elimination method. Learn how to eliminate variables and find the values of x and y in a step-by-step solution. Suitable for students in Grades 8-10.

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