Question

Plug in x=3 and x=-2 for both of the original equations to show that the equations can have more than one solution. Solve the equations. Explain your work. equation 2: 3x+3=7x+3. equation 6: 3(4x+8)=4(3x+6)

Answer 1

Let's analyze and solve the given equations step by step.

### Equation 2: \( 3x + 3 = 7x + 3 \)

1. **Substituting \( x = 3 \)**:
\[
3(3) + 3 = 7(3) + 3 \\
9 + 3 = 21 + 3 \\
12 = 24 \quad \text{(False)}
\]

2. **Substituting \( x = -2 \)**:
\[
3(-2) + 3 = 7(-2) + 3 \\
-6 + 3 = -14 + 3 \\
-3 = -11 \quad \text{(False)}
\]

Now, let's solve the equation for \( x \):

\[
3x + 3 = 7x + 3
\]

Subtract \( 3 \) from both sides:

\[
3x = 7x
\]

Subtract \( 7x \) from both sides:

\[
3x - 7x = 0 \\
-4x = 0
\]

Divide by \( -4 \):

\[
x = 0
\]

### Equation 6: \( 3(4x + 8) = 4(3x + 6) \)

1. **Substituting \( x = 3 \)**:
\[
3(4(3) + 8) = 4(3(3) + 6) \\
3(12 + 8) = 4(9 + 6) \\
3(20) = 4(15) \\
60 = 60 \quad \text{(True)}
\]

2. **Substituting \( x = -2 \)**:
\[
3(4(-2) + 8) = 4(3(-2) + 6) \\
3(-8 + 8) = 4(-6 + 6) \\
3(0) = 4(0) \\
0 = 0 \quad \text{(True)}
\]

Now, let's solve the equation for \( x \):

\[
3(4x + 8) = 4(3x + 6)
\]

Expanding both sides:

\[
12x + 24 = 12x + 24
\]

After simplifying, this equation reduces to:

\[
12x + 24 - 12x - 24 = 0 \\
0 = 0
\]

This indicates that the equation is an identity, meaning it is true for all values of \( x \), hence there are infinitely many solutions.

### Summary:

- **Equation 2 (\( 3x + 3 = 7x + 3 \)) has a single solution**: \( x = 0 \).
- **Equation 6 (\( 3(4x + 8) = 4(3x + 6) \)) has infinitely many solutions** since it simplifies to \( 0 = 0 \).

Thus, substituting \( x = 3 \) and \( x = -2 \) did not result in consistent true statements for the first equation, while both values satisfy the second equation, indicating multiple solutions.