Question

Plug in x=2 and x=-3 for both of the original equations to show that the equations
will have no solutions. Explain your work.
2x + 3 = 2x + 7
5x + 5 = 5x + 4

Answer 1

To determine if the two equations have solutions at \(x = 2\) and \(x = -3\), let's first evaluate both equations for these values.

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

1. **Plugging in \(x = 2\):**
\[
2(2) + 3 = 2(2) + 7
\]
\[
4 + 3 = 4 + 7
\]
\[
7 = 11 \quad \text{(Not True)}
\]

2. **Plugging in \(x = -3\):**
\[
2(-3) + 3 = 2(-3) + 7
\]
\[
-6 + 3 = -6 + 7
\]
\[
-3 = 1 \quad \text{(Not True)}
\]

### Second Equation: \(5x + 5 = 5x + 4\)

1. **Plugging in \(x = 2\):**
\[
5(2) + 5 = 5(2) + 4
\]
\[
10 + 5 = 10 + 4
\]
\[
15 = 14 \quad \text{(Not True)}
\]

2. **Plugging in \(x = -3\):**
\[
5(-3) + 5 = 5(-3) + 4
\]
\[
-15 + 5 = -15 + 4
\]
\[
-10 = -11 \quad \text{(Not True)}
\]

### Conclusion
For both equations, substituting \(x = 2\) or \(x = -3\) results in false statements. This shows that neither \(x = 2\) nor \(x = -3\) are solutions for either equation.

### Explanation
Moreover, let's analyze the structure of the equations:

1. The first equation simplifies to \(3 = 7\), indicating inconsistency, meaning there are no solutions.
2. The second equation simplifies to \(5 = 4\), indicating inconsistency as well, meaning it also has no solutions.

Both equations are indeed inconsistent (no values of \(x\) will satisfy them), confirming that they have no solutions.