Question

sort each equation by the number of solutions it has.
no solution
one solution
infinitely many solutions
10m - 5 = -5 + 10m
9x + 6 = 3(2 + 3x)
3x = x + 8

Explanation:

Step1: Analyze \( 3x = x + 8 \)

Subtract \( x \) from both sides: \( 3x - x = x + 8 - x \), which simplifies to \( 2x = 8 \).

Step2: Solve for \( x \)

Divide both sides by 2: \( \frac{2x}{2}=\frac{8}{2} \), so \( x = 4 \). This equation has one solution.

Now, check the other equations:

• For \( 10m - 5 = -5 + 10m \): Subtract \( 10m \) from both sides, we get \( -5=-5 \), which is always true, so infinitely many solutions.
• For \( 9x + 6 = 3(2 + 3x) \): Expand the right side: \( 9x + 6 = 6 + 9x \), subtract \( 9x \) from both sides, we get \( 6 = 6 \), which is always true, so infinitely many solutions.

Answer:

• No solution: (empty)
• One solution: \( 3x = x + 8 \)
• Infinitely many solutions: \( 10m - 5 = -5 + 10m \), \( 9x + 6 = 3(2 + 3x) \)