Solve the system of equations: (2x + 3y = 5) and (x - y = 2).

To solve the system of equations:

\[ 
\begin{align*}
1. \quad 2x + 3y &= 5 \\
2. \quad x - y &= 2
\end{align*}
\]

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

### Step 1: Eliminate \( y \)

To eliminate \( y \), we can multiply the second equation by 3 so that the coefficients of \( y \) in both equations are opposites:

\[
3(x - y) = 3 \cdot 2
\]

This gives us the new system:

\[ 
\begin{align*}
1. \quad 2x + 3y &= 5 \\
2. \quad 3x - 3y &= 6
\end{align*}
\]

### Step 2: Add the equations

Add the two equations together to eliminate \( y \):

\[
(2x + 3y) + (3x - 3y) = 5 + 6
\]

This simplifies to:

\[
5x = 11
\]

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

Divide both sides by 5 to find \( x \):

\[
x = \frac{11}{5}
\]

### Step 4: Substitute back to find \( y \)

Substitute \( x = \frac{11}{5} \) back into the second original equation to find \( y \):

\[
x - y = 2 \Rightarrow \frac{11}{5} - y = 2
\]

Solving for \( y \):

\[
-y = 2 - \frac{11}{5}
\]

Convert 2 to a fraction with a denominator of 5:

\[
2 = \frac{10}{5}
\]

So:

\[
-y = \frac{10}{5} - \frac{11}{5} = -\frac{1}{5}
\]

Thus:

\[
y = \frac{1}{5}
\]

### Solution

The solution to the system of equations is:

\[
x = \frac{11}{5}, \quad y = \frac{1}{5}
\]

Would you like further details or have any questions?

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Here are some additional related questions you might find useful:

1. What is another method to solve this system of equations?
2. How do you verify the solution of a system of equations?
3. Can the elimination method be used for nonlinear equations?
4. What are the advantages of using the substitution method?
5. How can systems of equations be solved graphically?
6. What happens if a system of equations has no solution?
7. How does the determinant of a matrix relate to solving systems of equations?
8. What is Cramer's Rule, and how is it used to solve systems of equations?

**Tip:** Always check your solution by substituting the values back into the original equations to ensure they satisfy both equations.

Learn how to solve the system of equations 2x + 3y = 5 and x - y = 2 with a detailed step-by-step solution using the elimination method. This problem covers key algebraic concepts, including systems of equations and the elimination method. Suitable for high school students in Grades 9-12.

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