Let's address each question one by one.
Question 1
We start with the equation: \[ 7x - 6 + 3 = 5x + 2x + 5. \]
First, simplify both sides of the equation: \[ 7x - 3 = 7x + 5. \]
Next, subtract \( 7x \) from both sides: \[ -3 = 5. \]
This statement is false, meaning that there are no values of \( x \) that can satisfy the equation. Therefore, the best description for the number of solutions for the given equation is: D: no solution.
Question 2
We need to identify which equation has infinitely many solutions. For an equation to have infinitely many solutions, it must be true for all values of the variable. This usually happens when simplifying the equation leads to a true statement (like \( 0 = 0 \)).
Let’s analyze each option:
A: \[ 9x - 3 = 3x + 6x + 2 + 3. \] Simplifying the right side: \[ 9x - 3 = 9x + 5. \] Subtract \( 9x \) from both sides: \[ -3 = 5, \] which is false (no solutions).
B: \[ 12(x + 8) = 11x - 5. \] Expanding both sides: \[ 12x + 96 = 11x - 5. \] Subtract \( 11x \) from both sides: \[ x + 96 = -5. \] This leads to \( x = -101 \) (one solution).
C: \[ 11x - 2x + 15 = 8 + 7 + 9x. \] This simplifies to: \[ 9x + 15 = 9x + 15. \] Here, we can subtract \( 9x \) from both sides: \[ 15 = 15, \] which is true for all \( x \) (infinitely many solutions).
D: \[ 5x - 8 = 11 - 7x + 12x. \] Simplifying that gives: \[ 5x - 8 = 5x + 11. \] Subtract \( 5x \) from both sides: \[ -8 = 11, \] which is false (no solutions).
The only equation that results in a true statement for all values of \( x \) is: C: 11x − 2x + 15 = 8 + 7 + 9x.
Thus, the answers are:
- D: no solution.
- C: 11x − 2x + 15 = 8 + 7 + 9x.