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.
7x + 3 = 7x + 3
3 (4x + 8) = 4 (3x + 6)
No special characters please

Answer 1

To determine if the two equations can have more than one solution, we will first evaluate both equations at \(x = 3\) and \(x = -2\), and then we will solve each equation.

### 1. Evaluating the Equations

#### Equation 1:
\(7x + 3 = 7x + 3\)

- At \(x = 3\):
- Left side: \(7(3) + 3 = 21 + 3 = 24\)
- Right side: \(7(3) + 3 = 21 + 3 = 24\)
- Both sides are equal: \(24 = 24\)

- At \(x = -2\):
- Left side: \(7(-2) + 3 = -14 + 3 = -11\)
- Right side: \(7(-2) + 3 = -14 + 3 = -11\)
- Both sides are equal: \(-11 = -11\)

Since both evaluations lead to true statements (the left side equals the right side), this equation is always true for any value of \(x\) because it is an identity. Thus, it has infinitely many solutions.

### 2. Solving Equation 2:
\(3(4x + 8) = 4(3x + 6)\)

First, we will expand both sides of the equation.

- Left side:
\[
3(4x + 8) = 12x + 24
\]

- Right side:
\[
4(3x + 6) = 12x + 24
\]

Now we have:
\[
12x + 24 = 12x + 24
\]

Again, both sides are equal, which means that this equation also represents an identity. Hence, it also has infinitely many solutions.

### Conclusion

Both equations \(7x + 3 = 7x + 3\) and \(3(4x + 8) = 4(3x + 6)\) are true for any value of \(x\) and therefore have infinitely many solutions.

When we plugged in \(x = 3\) and \(x = -2\), both resulted in true statements confirming that there is not a unique solution but rather an infinite number of solutions.