Solve the system of equations by the substitution method.
left brace Start 2 By 1 Matrix 1st Row 1st Column y equals 3 x plus 1 2nd Row 1st Column 3 y minus 4 x equals 13 EndMatrix y=3x+1	3y−4x=13

We are given the system of equations:

\[
\begin{aligned}
1. \ y &= 3x + 1 \\
2. \ 3y - 4x &= 13
\end{aligned}
\]

We'll solve this system using the **substitution method**, where we substitute the expression for \( y \) from the first equation into the second equation.

### Step 1: Substitute \( y = 3x + 1 \) into the second equation

Take the second equation:

\[
3y - 4x = 13
\]

Substitute \( y = 3x + 1 \) into this equation:

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

### Step 2: Simplify the equation

Distribute the \( 3 \) on the left-hand side:

\[
9x + 3 - 4x = 13
\]

Combine like terms:

\[
(9x - 4x) + 3 = 13 \quad \Rightarrow \quad 5x + 3 = 13
\]

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

Subtract 3 from both sides:

\[
5x = 10
\]

Now, divide both sides by 5:

\[
x = 2
\]

### Step 4: Substitute \( x = 2 \) back into the first equation

Now that we know \( x = 2 \), substitute this into the first equation to find \( y \):

\[
y = 3x + 1 = 3(2) + 1 = 6 + 1 = 7
\]

### Final Answer

The solution to the system of equations is:

\[
x = 2, \quad y = 7
\]

### Verification

To verify, substitute \( x = 2 \) and \( y = 7 \) back into the second equation:

\[
3y - 4x = 13 \quad \Rightarrow \quad 3(7) - 4(2) = 21 - 8 = 13
\]

Both equations are satisfied, so the solution is correct.

---
Would you like further details or have any questions?

Here are 5 related questions to extend your understanding:

1. How would you solve the system if the first equation were \( y = 3x - 1 \)?
2. Can you explain what would happen if the two equations resulted in no solution?
3. What would you do if both equations represented the same line (infinite solutions)?
4. How would you graphically represent this system of equations?
5. How does the substitution method compare with the elimination method?

**Tip:** Always check your solution by substituting the values back into the original equations.

Solve the system of equations by the substitution method. y = 3x + 1, 3y - 4x = 13

Learn how to solve the system of equations y = 3x + 1 and 3y - 4x = 13 using the substitution method. This problem covers key algebraic concepts and demonstrates a step-by-step approach to solving linear equations, suitable for students in Grades 8-10.

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