Question

2x+3= 2x+7 5x+5=5x+4 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.

Answer 1

To show that the equations \(2x + 3 = 2x + 7\) and \(5x + 5 = 5x + 4\) have no solutions, we can analyze each equation separately.

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

1. **Simplify the Equation:**
Subtract \(2x\) from both sides:
\[
2x + 3 - 2x = 2x + 7 - 2x
\]
This simplifies to:
\[
3 = 7
\]
This is a contradiction, as \(3\) does not equal \(7\). Therefore, this equation has no solutions.

2. **Plugging in Values:**
- **For \(x = 2\)**:
\[
2(2) + 3 = 4 + 3 = 7
\]
\[
2(2) + 7 = 4 + 7 = 11
\]
Since \(7 \neq 11\), the equation does not hold.

- **For \(x = -3\)**:
\[
2(-3) + 3 = -6 + 3 = -3
\]
\[
2(-3) + 7 = -6 + 7 = 1
\]
Since \(-3 \neq 1\), the equation does not hold.

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

1. **Simplify the Equation:**
Subtract \(5x\) from both sides:
\[
5x + 5 - 5x = 5x + 4 - 5x
\]
This simplifies to:
\[
5 = 4
\]
This is also a contradiction, as \(5\) does not equal \(4\). Therefore, this equation has no solutions.

2. **Plugging in Values:**
- **For \(x = 2\)**:
\[
5(2) + 5 = 10 + 5 = 15
\]
\[
5(2) + 4 = 10 + 4 = 14
\]
Since \(15 \neq 14\), the equation does not hold.

- **For \(x = -3\)**:
\[
5(-3) + 5 = -15 + 5 = -10
\]
\[
5(-3) + 4 = -15 + 4 = -11
\]
Since \(-10 \neq -11\), the equation does not hold.

### Conclusion

Both equations \(2x + 3 = 2x + 7\) and \(5x + 5 = 5x + 4\) lead to contradictions when simplified, indicating that they have no solutions. Plugging in specific values \(x = 2\) and \(x = -3\) for both equations confirms that neither equation holds true for those values, further illustrating that there are no solutions.