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Let's solve the system of equations using the **substitution method**.

The system of equations is:

1. \( 2x - 3y = 5 \)
2. \( 4x + y = 11 \)

### Step-by-Step Solution:

**Step 1: Solve one equation for one variable.**

From the second equation \( 4x + y = 11 \), solve for \( y \):

\[
y = 11 - 4x
\]

**Step 2: Substitute the expression for \( y \) into the first equation.**

Now substitute \( y = 11 - 4x \) into the first equation \( 2x - 3y = 5 \):

\[
2x - 3(11 - 4x) = 5
\]

Simplify the equation:

\[
2x - 33 + 12x = 5
\]

Combine like terms:

\[
14x - 33 = 5
\]

**Step 3: Solve for \( x \).**

Add 33 to both sides:

\[
14x = 38
\]

Now, divide by 14:

\[
x = \frac{38}{14} = \frac{19}{7}
\]

**Step 4: Substitute \( x = \frac{19}{7} \) back into the expression for \( y \).**

Substitute into \( y = 11 - 4x \):

\[
y = 11 - 4\left(\frac{19}{7}\right) = 11 - \frac{76}{7}
\]

Convert 11 into a fraction with denominator 7:

\[
y = \frac{77}{7} - \frac{76}{7} = \frac{1}{7}
\]

**Final Solution:**

\[
x = \frac{19}{7}, \quad y = \frac{1}{7}
\]

Would you like any more details or clarification on the steps? 

Here are 5 related questions to deepen your understanding:

1. How can we solve the same system using the elimination method?
2. What are the advantages of using the substitution method for this problem?
3. Can this system of equations be solved graphically?
4. How can you verify that the solution \( \left( \frac{19}{7}, \frac{1}{7} \right) \) is correct?
5. How does solving for one variable first help in the substitution method?

**Tip:** Always simplify the equation fully before solving for the unknown variable to avoid mistakes.

Solve the system of equations using the substitution method: 2x - 3y = 5 and 4x + y = 11.

Learn how to solve a system of linear equations, 2x - 3y = 5 and 4x + y = 11, using the substitution method. This detailed solution walks through solving for one variable and substituting it into the other equation. Suitable for students in Grades 9-11.

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