\begin{aligned}2x-3y=-1
\\5x+2y=7\end{aligned}

To solve the system of equations:

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

we can use either substitution or elimination. Here, I'll use the elimination method:

### Step 1: Multiply the equations to align coefficients

To eliminate one variable, we need to manipulate the equations. Let's eliminate \( y \) by making the coefficients of \( y \) equal. 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

Now, add the two equations to eliminate \( y \):

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

which simplifies to:

\[
19x = 19.
\]

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

Divide both sides by 19:

\[
x = 1.
\]

### Step 4: Substitute \( x \) into one of the original equations

Use 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 to the system is:

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

Would you like more details or have any questions?

### Related Questions

1. How can we solve a system of equations using substitution?
2. What is the graphical interpretation of the solution to this system?
3. How would the solution change if the equations were dependent or inconsistent?
4. Can we solve this system using matrix methods like Gaussian elimination?
5. What are the applications of solving systems of equations in real-world problems?

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

Learn how to solve the system of equations 2x - 3y = -1 and 5x + 2y = 7 step-by-step using the elimination method. This problem involves systems of linear equations and is suitable for high school students.

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