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The system of equations provided is:

\[
\begin{aligned}
1) &\ 2x - 3y = -5 \\
2) &\ 3x + 4y = 1
\end{aligned}
\]

To solve this using **elimination**, we will eliminate one variable (either \(x\) or \(y\)) by multiplying both equations to align the coefficients for one variable and subtract or add them to eliminate that variable. Let's eliminate \(x\).

### Step 1: Multiply to align coefficients of \(x\)
We want the coefficients of \(x\) in both equations to be the same, so we'll multiply the first equation by 3 and the second equation by 2:

\[
\text{Equation 1} \times 3: \quad 6x - 9y = -15
\]
\[
\text{Equation 2} \times 2: \quad 6x + 8y = 2
\]

### Step 2: Subtract the equations
Now, subtract the second equation from the first to eliminate \(x\):

\[
(6x - 9y) - (6x + 8y) = -15 - 2
\]
This simplifies to:
\[
-17y = -17
\]

### Step 3: Solve for \(y\)
Divide both sides by \(-17\):
\[
y = 1
\]

### Step 4: Substitute \(y = 1\) back into one of the original equations
We'll substitute \(y = 1\) into the first equation \(2x - 3y = -5\):

\[
2x - 3(1) = -5
\]
\[
2x - 3 = -5
\]
\[
2x = -2
\]
\[
x = -1
\]

### Final Solution:
The solution set is \((x, y) = (-1, 1)\).

Would you like further details or clarifications?

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Here are some follow-up questions related to this topic:
1. What other methods could be used to solve this system of equations?
2. How would the solution change if the system were inconsistent?
3. How can you check if the solution is correct using substitution?
4. Can you solve the same system by substitution instead of elimination?
5. What happens if both equations are multiples of each other?

**Tip:** When using elimination, it’s helpful to focus on aligning the coefficients of one variable as early as possible for efficiency.

Find the solution set of the system of linear equations NOT by graphing.

2x - 3y = -5
3x + 4y = 1

Learn how to solve the system of linear equations 2x - 3y = -5 and 3x + 4y = 1 using the elimination method. This problem demonstrates step-by-step solutions, including aligning coefficients and eliminating variables. Suitable for students in Grades 9-11.

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