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Let's solve the given system of equations by substitution. The system is:

\[
\begin{aligned}
1. \ y &= 4x + 1 \\
2. \ 2y - 4x &= 14
\end{aligned}
\]

### Step-by-Step Solution:

1. **Substitute the expression for \(y\) from Equation 1 into Equation 2:**

Since \(y = 4x + 1\), substitute this into the second equation:

\[
2(4x + 1) - 4x = 14
\]

2. **Simplify the equation:**

Expand the left-hand side:

\[
2(4x + 1) = 8x + 2
\]

So the equation becomes:

\[
8x + 2 - 4x = 14
\]

3. **Combine like terms:**

\[
4x + 2 = 14
\]

4. **Solve for \(x\):**

Subtract 2 from both sides:

\[
4x = 12
\]

Now, divide by 4:

\[
x = 3
\]

5. **Substitute \(x = 3\) back into Equation 1 to find \(y\):**

\[
y = 4(3) + 1 = 12 + 1 = 13
\]

### Solution:
The solution to the system is \(x = 3\) and \(y = 13\).

Would you like more details or have any questions?

### Relative Questions:
1. How would you solve the system if you used elimination instead of substitution?
2. Can substitution work on all systems of equations? Why or why not?
3. How can graphing help in solving this system of equations?
4. What would happen if the system had no solutions or infinite solutions?
5. How does substitution differ from other algebraic methods like elimination?

### Tip:
Always simplify equations fully before substituting values back into the system to avoid errors.

Solve the system of equations by the substitution method:
1. y = 4x + 1
2. 2y - 4x = 14

Learn how to solve a system of linear equations using the substitution method. This example involves equations y = 4x + 1 and 2y - 4x = 14. Detailed, step-by-step solution ideal for students in Grades 8-10.

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