Linear Equations in One Variable
Consider the equation 7x − 6 + 3 = 5x + 2x + 5.
Question 1
Which BEST describes the number of solutions for the given equation?
Responses
A one solutionone solution
B two solutionstwo solutions
C infinitely many solutionsinfinitely many solutions
D no solutionno solution
Question 2
Which equation has infinitely many solutions?
Responses
A 9x − 3 = 3x + 6x + 2 + 39x − 3 = 3x + 6x + 2 + 3
B 12(x + 8) = 11x − 512(x + 8) = 11x − 5
C 11x − 2x + 15 = 8 + 7 + 9x11x − 2x + 15 = 8 + 7 + 9x
D 5x − 8 = 11 − 7x + 12x
Question 1: The given equation has one solution. Therefore, the correct answer is A) one solution.
Question 2: The equation that has infinitely many solutions is C) 11x − 2x + 15 = 8 + 7 + 9x.
are you sure
I apologize for the confusion in my previous response. Let's reevaluate the equations.
Question 1: By simplifying the equation 7x − 6 + 3 = 5x + 2x + 5, we get 7x - 3 = 7x + 5. When we further simplify it, we get -3 = 5 which is not true. This means that there are no solutions for the given equation. Therefore, the correct answer is D) no solution.
Question 2: Let's evaluate the given options:
A) 9x − 3 = 3x + 6x + 2 + 3: This equation can be simplified to 9x - 3 = 9x + 5, which is not true. Therefore, it does not have infinitely many solutions.
B) 12(x + 8) = 11x − 5: This equation can be simplified to 12x + 96 = 11x - 5. By further simplification, we get x = -101. This is a specific value for x, so it does not have infinitely many solutions.
C) 11x − 2x + 15 = 8 + 7 + 9x: This equation can be simplified to 9x + 15 = 15 + 9x, which is true for all values of x. This means that the equation has infinitely many solutions.
D) 5x − 8 = 11 − 7x + 12x: This equation can be simplified to 5x - 8 = 11 + 5x, which is not true. Therefore, it does not have infinitely many solutions.
Therefore, the correct answer is C) 11x − 2x + 15 = 8 + 7 + 9x, which has infinitely many solutions.
Question 1: To determine the number of solutions for the equation 7x − 6 + 3 = 5x + 2x + 5, we need to simplify it.
7x − 6 + 3 = 5x + 2x + 5
First, we can combine like terms on both sides of the equation:
7x - 3 = 7x + 5
Next, we can simplify further by subtracting 7x from both sides:
-3 = 5
Since -3 is not equal to 5, the equation is contradicting itself, meaning there is no value of x that can satisfy this equation.
Therefore, the answer to Question 1 is D) no solution.
Question 2: To determine the equation that has infinitely many solutions, we need to examine each option.
A) 9x − 3 = 3x + 6x + 2 + 3
First, let's simplify this equation by combining like terms:
9x - 3 = 9x + 5
Subtracting 9x from both sides:
-3 = 5
This equation contradicts itself and has no solution. So option A does not have infinitely many solutions.
B) 12(x + 8) = 11x − 5
Let's simplify this equation:
12x + 96 = 11x - 5
Subtracting 11x from both sides:
x + 96 = -5
Subtracting 96 from both sides:
x = -101
This equation has a single solution, so option B does not have infinitely many solutions.
C) 11x − 2x + 15 = 8 + 7 + 9x
Let's simplify this equation:
9x + 15 = 15 + 9x
Subtracting 9x from both sides:
15 = 15
This equation is always true, regardless of the value of x. Therefore, it has infinitely many solutions.
Therefore, the answer to Question 2 is C) 11x − 2x + 15 = 8 + 7 + 9x.