\begin{aligned}2x-3y=-1

\\5x+2y=7\end{aligned}

Let's solve the system of equations:

\[
\begin{aligned}
2x - 3y &= -1, \\
5x + 2y &= 7.
\end{aligned}
\]

We will use the elimination method:

### Step 1: Eliminate one variable

To eliminate \( y \), we'll align the coefficients by multiplying the equations appropriately:

Multiply the first equation by 2 and the second equation by 3:

\[
\begin{aligned}
2(2x - 3y) &= 2(-1), \\
3(5x + 2y) &= 3(7).
\end{aligned}
\]

Simplify these equations:

\[
\begin{aligned}
4x - 6y &= -2, \\
15x + 6y &= 21.
\end{aligned}
\]

### Step 2: Add the equations to eliminate \( y \)

\[
(4x - 6y) + (15x + 6y) = -2 + 21,
\]

which simplifies to:

\[
19x = 19.
\]

### Step 3: Solve for \( x \)

\[
x = \frac{19}{19} = 1.
\]

### Step 4: Substitute \( x = 1 \) into the first equation

\[
2(1) - 3y = -1.
\]

Simplify:

\[
2 - 3y = -1.
\]

Subtract 2 from both sides:

\[
-3y = -3.
\]

Divide by -3:

\[
y = 1.
\]

### Solution

The solution is:

\[
x = 1, \quad y = 1.
\]

Would you like more details or have any questions?

### Related Questions

1. What if one equation is multiplied incorrectly? How does it affect the solution?
2. How do you solve using substitution instead of elimination?
3. What happens if there are no solutions (inconsistent system)?
4. How can we use a graphing method to verify this solution?
5. How does matrix algebra help in solving this system?

**Tip:** Double-check your steps by plugging values back into both equations to ensure accuracy!

Learn how to solve the system of equations 2x - 3y = -1 and 5x + 2y = 7 using the elimination method with a step-by-step solution. This problem covers linear equations and the elimination method. Suitable for high school students (Grades 9-12).

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