To solve the system of linear equations:

1. \(2x + 3y = 14\)
2. \(3x + 2y = 8\)

we can use either the substitution method or the elimination method. Here, I will use the elimination method to solve the system.

### Step-by-Step Solution

**Step 1: Align the equations**

\[
2x + 3y = 14 \quad \text{(Equation 1)}
\]
\[
3x + 2y = 8 \quad \text{(Equation 2)}
\]

**Step 2: Multiply the equations to align coefficients**

To eliminate \(x\), we can multiply Equation 1 by 3 and Equation 2 by 2:

\[
3(2x + 3y) = 3 \cdot 14 \implies 6x + 9y = 42
\]
\[
2(3x + 2y) = 2 \cdot 8 \implies 6x + 4y = 16
\]

Now we have the system:

\[
6x + 9y = 42 \quad \text{(Equation 3)}
\]
\[
6x + 4y = 16 \quad \text{(Equation 4)}
\]

**Step 3: Subtract the equations to eliminate \(x\)**

Subtract Equation 4 from Equation 3:

\[
(6x + 9y) - (6x + 4y) = 42 - 16
\]
\[
5y = 26
\]

**Step 4: Solve for \(y\)**

\[
y = \frac{26}{5} = 5.2
\]

**Step 5: Substitute \(y\) back into one of the original equations**

Let's substitute \(y = 5.2\) into Equation 1:

\[
2x + 3(5.2) = 14
\]
\[
2x + 15.6 = 14
\]
\[
2x = 14 - 15.6
\]
\[
2x = -1.6
\]
\[
x = \frac{-1.6}{2} = -0.8
\]

**Solution:**

\[
x = -0.8, \quad y = 5.2
\]

So, the solution to the system of equations is:

\[
(x, y) = (-0.8, 5.2)
\]

Would you like more details on any of these steps, or do you have any questions?

### 5 Related Questions:

1. How can we verify the solution to this system of equations?
2. What is the graphical interpretation of solving a system of linear equations?
3. How can we solve this system using the substitution method?
4. What are some real-world applications of solving systems of equations?
5. How can we solve systems of equations with more than two variables?

**Tip:** When solving systems of equations, it's helpful to 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 linear equations 2x + 3y = 14 and 3x + 2y = 8 using the elimination method. This detailed step-by-step solution explains key algebraic concepts, suitable for students in Grades 8-10.

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