Let's analyze the questions one at a time.
Question 1: We need to find the number of solutions for the equation \( 7x - 6 + 3 = 5x + 2x + 5 \).
First, simplify both sides of the equation:
-
Left side:
\( 7x - 6 + 3 = 7x - 3 \) -
Right side:
\( 5x + 2x + 5 = 7x + 5 \)
This gives us the equation:
\[ 7x - 3 = 7x + 5 \]
Now, let's bring all terms involving \( x \) to one side and constants to the other:
Subtract \( 7x \) from both sides:
\[ -3 = 5 \]
Since \(-3\) does not equal \(5\), the equation is a contradiction, meaning there are no solutions for the given equation.
Answer to Question 1: D) no solution
Question 2: We need to find which of the following equations has infinitely many solutions. An equation has infinitely many solutions if it can be simplified to the form \( 0 = 0 \).
Let's evaluate each option:
A) \( 9x − 3 = 3x + 6x + 2 + 3 \)
Simplifying the right side:
\( 3x + 6x + 2 + 3 = 9x + 5 \)
The equation becomes:
\( 9x - 3 = 9x + 5 \)
This simplifies to \(-3 = 5\), which is not true. No solution.
B) \( 12(x + 8) = 11x - 5 \)
Expanding the left side:
\( 12x + 96 = 11x - 5 \)
Moving terms gives us:
\( 12x - 11x = -5 - 96 \) → \( x = -101 \). One solution.
C) \( 11x - 2x + 15 = 8 + 7 + 9x \)
Simplifying both sides:
Left side: \( 9x + 15 \)
Right side: \( 15 + 9x \)
This simplifies to \( 9x + 15 = 9x + 15 \), which is always true. Infinitely many solutions.
D) \( 5x − 8 = 11 − 7x + 12x \)
Simplifying the right side:
\( 5x - 8 = 11 + 5x \)
Moving terms gives us:
\( -8 = 11\), which is not true. No solution.
Answer to Question 2: C) \( 11x - 2x + 15 = 8 + 7 + 9x \) has infinitely many solutions.