Question

which of the following equations has no real solutions?
a. 5x - 7 = 35x - 7
b. 5x - 7 = 5x - 7
c. 5(x + 7) = 5(x + 9)

Explanation:

Step1: Analyze Option A

Start with the equation \(5x - 7 = 35x - 7\). Subtract \(5x\) from both sides: \(-7 = 30x - 7\). Then add 7 to both sides: \(0 = 30x\), so \(x = 0\). This has a real solution.

Step2: Analyze Option B

Take the equation \(5x - 7 = 5x - 7\). Subtract \(5x\) from both sides: \(-7 = -7\). This is an identity, so there are infinitely many real solutions.

Step3: Analyze Option C

Start with \(5(x + 7) = 5(x + 9)\). Distribute both sides: \(5x + 35 = 5x + 45\). Subtract \(5x\) from both sides: \(35 = 45\), which is false. So this equation has no real solutions.

Answer:

C. \(5(x + 7) = 5(x + 9)\)