Find the solution set of the given equations using elimination: 2x - 5y = -13; -3y + 4 = -5x

We are tasked with solving the system of equations using the **elimination method**:

1. \( 2x - 5y = -13 \) \[ \text{(Equation 1)} \]
2. \( -3y + 4 = -5x \) \[ \text{(Equation 2)} \]

### Step 1: Rearrange the second equation
To align both equations better, we need to rearrange Equation 2 into a form similar to Equation 1:

\[
-3y + 4 = -5x
\]
Add \( 5x \) to both sides:
\[
5x - 3y = 4 \quad \text{(Equation 3)}
\]

Now, we have the system:

\[
2x - 5y = -13 \quad \text{(Equation 1)}
\]
\[
5x - 3y = 4 \quad \text{(Equation 3)}
\]

### Step 2: Multiply to eliminate one variable
We will eliminate \( x \). To do that, let's multiply the first equation by 5 and the second equation by 2, so that the coefficients of \( x \) will be the same:

Multiply Equation 1 by 5:
\[
(5)(2x - 5y) = 5(-13) \quad \Rightarrow \quad 10x - 25y = -65 \quad \text{(Equation 4)}
\]

Multiply Equation 3 by 2:
\[
(2)(5x - 3y) = (2)(4) \quad \Rightarrow \quad 10x - 6y = 8 \quad \text{(Equation 5)}
\]

### Step 3: Subtract the equations
Now subtract Equation 5 from Equation 4 to eliminate \( x \):

\[
(10x - 25y) - (10x - 6y) = (-65) - (8)
\]
This simplifies to:
\[
10x - 25y - 10x + 6y = -65 - 8
\]
\[
-19y = -73
\]
Solving for \( y \):
\[
y = \frac{-73}{-19} =  \frac{73}{19} = 3.84 \quad \text{(rounded to 2 decimal places)}.
\]

### Step 4: Substitute \( y \) back into one of the original equations
Let's substitute \( y = 3.84 \) into Equation 1 to solve for \( x \):

\[
2x - 5(3.84) = -13
\]
\[
2x - 19.2 = -13
\]
Add 19.2 to both sides:
\[
2x = 6.2
\]
Solving for \( x \):
\[
x = \frac{6.2}{2} = 3.1.
\]

### Final Solution Set:
\[
x = 3.1, \quad y = 3.84.
\]

This is the solution set for the system of equations.

Would you like further details or clarifications on any steps?

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**Related Questions:**
1. How does the substitution method compare to elimination?
2. Can you solve the same system graphically?
3. What happens when a system of equations has no solution?
4. How do you solve systems of three variables using elimination?
5. What are other methods for solving systems of linear equations?

**Tip:** Always check your solution by substituting it back into the original equations.

Learn how to solve the system of equations 2x - 5y = -13 and -3y + 4 = -5x using the elimination method. This detailed solution covers key algebraic concepts like linear equations, the elimination method, and properties of equality. Suitable for students in Grades 8-10.

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