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
To begin, we can simplify the equation given on the left side:
7x - 6 + 3 = 5x + 2x + 5
7x - 3 = 7x + 5
Next, we can subtract 7x from both sides to eliminate the variable:
-3 = 5
Since -3 does not equal 5, this means that there are no solutions for the given equation.
Therefore, the answer to Question 1 is D) no solution.
Question 2
To determine which equation has infinitely many solutions, we need to find an equation in which the variable cancels out and we are left with a true statement.
Looking at the equation options, we can simplify each one:
A) 9x - 3 = 3x + 6x + 2 + 3
Simplifying this equation, we get:
9x - 3 = 9x + 5
We can subtract 9x from both sides:
-3 = 5
This equation is not true, so it does not have infinitely many solutions.
B) 12(x + 8) = 11x - 5
Expanding the left side of the equation, we get:
12x + 96 = 11x - 5
Simplifying further, we can subtract 11x from both sides:
12x - 11x + 96 = -5
This simplifies to:
x + 96 = -5
Subtracting 96 from both sides, we get:
x = -101
This equation has only one solution, so it does not have infinitely many solutions.
C) 11x - 2x + 15 = 8 + 7 + 9x
Simplifying this equation, we get:
9x + 15 = 15 + 9x
Both sides of the equation are the same, so this equation has infinitely many solutions. This is because any value of x will make the equation true.
D) 5x - 8 = 11 - 7x + 12x
Combining like terms, we get:
5x - 8 = 11 + 5x
Subtracting 5x from both sides, we get:
-8 = 11
This equation is not true, so it does not have infinitely many solutions.
Therefore, the answer to Question 2 is C) 11x - 2x + 15 = 8 + 7 + 9x.
the other one
I apologize for the confusion. Let's analyze the options once again:
Question 2:
A) 9x − 3 = 3x + 6x + 2 + 3
Simplifying this equation, we get:
9x - 3 = 9x + 5
Subtracting 9x from both sides, we have:
-3 = 5
This equation is not true, so it does not have infinitely many solutions.
B) 12(x + 8) = 11x − 5
Expanding the left side of the equation, we get:
12x + 96 = 11x - 5
Subtracting 11x from both sides, we have:
12x - 11x + 96 = -5
Simplifying further, we get:
x + 96 = -5
Subtracting 96 from both sides, we get:
x = -101
This equation has only one solution, so it does not have infinitely many solutions.
C) 11x − 2x + 15 = 8 + 7 + 9x
Simplifying this equation, we get:
9x + 15 = 15 + 9x
Both sides of the equation are the same, so this equation has infinitely many solutions. This is because any value of x will make the equation true.
D) 5x − 8 = 11 − 7x + 12x
Combining like terms, we get:
17x - 8 = 11
Adding 8 to both sides, we have:
17x = 19
Dividing by 17, we find:
x = 19/17
This equation has only one solution, so it does not have infinitely many solutions.
Therefore, the answer to Question 2 is C) 11x − 2x + 15 = 8 + 7 + 9x.
To solve the equation 7x − 6 + 3 = 5x + 2x + 5, we can simplify both sides of the equation.
Step 1: Combine like terms on the left side
Combining like terms on the left side, we have:
7x − 6 + 3 = 7x − 3
Step 2: Combine like terms on the right side
Combining like terms on the right side, we have:
5x + 2x + 5 = 7x + 5
Step 3: Simplify both sides
Our equation now becomes:
7x − 3 = 7x + 5
Step 4: Solve for x
To solve for x, we want to isolate the x term on one side of the equation. However, in this case, we can see that the x terms on both sides cancel each other out.
Since the equation 7x − 3 = 7x + 5 has no x terms remaining, we can conclude that this equation has no solution.
Therefore, the answer to Question 1 is D: no solution.
Now, let's consider Question 2 which asks for an equation with infinitely many solutions.
An equation has infinitely many solutions if it is an identity, meaning both sides of the equation are always equal, regardless of the value of x.
Among the given options, the equation that has infinitely many solutions is:
Option A: 9x − 3 = 3x + 6x + 2 + 3
To determine if this equation has infinitely many solutions, we need to simplify it:
9x − 3 = 9x + 2 + 3
Now, combining like terms:
9x − 3 = 9x + 5
As we can see, both sides of the equation have the same x term and a constant term. Therefore, they will always be equal, regardless of the value of x. This means the equation has infinitely many solutions.
Therefore, the answer to Question 2 is A: 9x − 3 = 3x + 6x + 2 + 3.
To solve these questions, we need to simplify the given equations and compare the coefficients of the variables.
Question 1:
We have the equation 7x − 6 + 3 = 5x + 2x + 5.
Simplifying both sides, we get 7x - 3 = 7x + 5.
Now, let's collect the terms with x on one side of the equation:
7x - 7x = 5 + 3.
0 = 8.
This equation is inconsistent because 0 cannot equal to 8. Therefore, there are no solutions (D) for this equation.
Question 2:
We need to check each equation to determine if it has infinitely many solutions.
A. 9x - 3 = 3x + 6x + 2 + 3
Simplifying both sides, we get 9x - 3 = 9x + 5.
Collecting the terms with x on one side, we have:
9x - 9x = 5 + 3.
0 = 8.
Similar to Question 1, this equation is inconsistent because 0 cannot equal to 8. So, this equation does not have infinitely many solutions.
B. 12(x + 8) = 11x - 5
Distributing 12 on the left side, we get: 12x + 96 = 11x - 5.
Collecting the terms with x on one side, we have:
12x - 11x = -5 - 96.
x = -101.
This equation has only one solution since x is equal to -101.
C. 11x - 2x + 15 = 8 + 7 + 9x
Simplifying both sides, we get 9x + 15 = 15 + 16x.
Collecting the terms with x on one side, we have:
9x - 16x = 15 - 15.
-7x = 0.
x = 0.
This equation also has only one solution since x is equal to 0.
D. 5x - 8 = 11 - 7x + 12x
Simplifying both sides, we get 5x - 8 = -7x + 11.
Collecting the terms with x on one side, we have:
5x + 7x - 12x = 11 + 8.
0 = 19.
Again, similar to Question 1, this equation is inconsistent because 0 cannot equal to 19. So, this equation does not have infinitely many solutions.
Therefore, the equation that has infinitely many solutions is A (9x - 3 = 3x + 6x + 2 + 3).