Solve the system of equations.

y = 4x+1

3x + 2y = 13

We are given the following system of equations:

1. \( y = 4x + 1 \)
2. \( 3x + 2y = 13 \)

### Step 1: Substitute \( y \) from the first equation into the second equation.

Since \( y = 4x + 1 \), substitute this into the second equation \( 3x + 2y = 13 \):

\[
3x + 2(4x + 1) = 13
\]

### Step 2: Simplify the equation.

Distribute the \( 2 \) on the left-hand side:

\[
3x + 8x + 2 = 13
\]

Combine like terms:

\[
11x + 2 = 13
\]

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

Subtract 2 from both sides:

\[
11x = 11
\]

Now, divide by 11:

\[
x = 1
\]

### Step 4: Substitute \( x = 1 \) back into the first equation to find \( y \).

Using \( y = 4x + 1 \):

\[
y = 4(1) + 1 = 5
\]

### Final Answer:
The solution to the system of equations is \( x = 1 \) and \( y = 5 \).

Would you like more details, or have any questions?

Here are 5 related questions for further exploration:

1. What would happen if the coefficient of \( y \) in the second equation changed?
2. Can you solve the system using the substitution method in a different way?
3. How would you approach solving the system graphically?
4. What are some examples of systems of equations that have no solutions?
5. Can you solve a system of equations using matrix methods?

**Tip:** When solving systems of equations, substitution is often most useful when one equation is already solved for a variable, while elimination is better when variables are aligned for easy elimination.

Solve the system of equations. y = 4x + 1 and 3x + 2y = 13

Learn how to solve the system of linear equations y = 4x + 1 and 3x + 2y = 13 using the substitution method. This problem covers key algebraic concepts such as solving systems of equations, simplifying expressions, and finding solutions for x and y. Ideal for students in Grades 8-10.

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