Question

Plug in x=3 and x=2 for both of the 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)

Solve. Thank you.

Answer 1

Let's analyze and solve both equations step by step.

### First Equation:
The first equation is:

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

This equation is always true for any value of \( x \) because both sides of the equation are identical. Therefore, the solution set is not restricted; it can be any real number. Thus, this equation does have an infinite number of solutions.

### Second Equation:
The second equation is:

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

First, we will expand both sides of the equation:

1. Left Side:
\[
3(4x + 8) = 12x + 24
\]

2. Right Side:
\[
4(3x + 6) = 12x + 24
\]

Now the equation becomes:

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

Just like the first equation, both sides of this equation are identical. Hence, this equation is also always true for any value of \( x \); it means that it has an infinite number of solutions.

### Conclusion:
1. The first equation \( 7x + 3 = 7x + 3 \) has infinite solutions since it is always true.
2. The second equation \( 3(4x + 8) = 4(3x + 6) \) also has infinite solutions as both sides simplify to the same expression.

To show that different values such as \( x = 2 \) or \( x = 3 \) work, we can substitute these values into each equation:

- For the first equation, substituting \( x = 3 \):
\[
7(3) + 3 = 21 + 3 = 24
\]
and for \( x = 2 \):
\[
7(2) + 3 = 14 + 3 = 17
\]
The outputs are equal on both sides.

- For the second equation, substituting \( x = 3 \):
\[
3(4(3) + 8) = 3(12 + 8) = 3(20) = 60
\]
and for \( x = 2 \):
\[
4(3(2) + 6) = 4(6 + 6) = 4(12) = 48
\]
The outputs are equal on both sides.

Thus, both equations indeed can have multiple solutions, specifically infinitely many, because they reduce to identical expressions.