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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

Answer 1

[AI-generated answer — Bot GPT 3.5]

Question 1:

The given equation is 7x − 6 + 3 = 5x + 2x + 5.
To solve this equation, we can simplify both sides by combining like terms:
7x - 3 = 7x + 5
Next, we can subtract 7x from both sides to eliminate the variable:
-3 = 5
Since this leads to a contradiction, there are no solutions for the given equation.
Therefore, the correct answer is D) no solution.

Question 2:
To determine the equation with infinitely many solutions, we need to find an equation where any value for the variable satisfies the equation.
Looking at the given options, we can see that option A) 9x − 3 = 3x + 6x + 2 + 39x − 3 = 3x + 6x + 2 + 3 can be simplified as follows:
9x - 3 = 9x + 5
By subtracting 9x from both sides, we get:
-3 = 5
Similar to the previous question, this leads to a contradiction and there are no solutions for this equation. Therefore, option A) does not have infinitely many solutions.

None of the other options seem to have the same pattern where any value for the variable satisfies the equation. Hence, the correct answer is none of the given options.

Answer 2

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

7x − 6 + 3 = 5x + 2x + 5

Answer 3

[AI-generated answer — Bot GPT 3.5]

To determine the number of solutions for the given equation, we can simplify it:

7x − 6 + 3 = 5x + 2x + 5

By combining like terms, we get:

7x - 3 = 7x + 5

Next, we can eliminate the variable by subtracting 7x from both sides:

-3 = 5

This leads to a contradiction, as -3 is not equal to 5. Therefore, there are no solutions for the given equation.

The correct answer is D) no solution.

Answer 4

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

Answer 5

[AI-generated answer — Bot GPT 3.5]

To determine which equation has infinitely many solutions, we need to find an equation where any value for the variable satisfies the equation.

Looking at the given options, we can see that option C) 11x − 2x + 15 = 8 + 7 + 9x11x − 2x + 15 = 8 + 7 + 9x can be simplified as follows:

9x + 15 = 15 + 9x

By subtracting 9x from both sides, we get:

15 = 15

This equation is true for any value of x, meaning that there are infinitely many solutions. Therefore, the correct answer is C) 11x − 2x + 15 = 8 + 7 + 9x11x − 2x + 15 = 8 + 7 + 9x.

Answer 6

how mny solutions 4(x + 5) - 5 = 8x+182

Answer 7

[AI-generated answer — Bot GPT 3.5]

To determine the number of solutions for the given equation, we can simplify and solve it:

4(x + 5) - 5 = 8x + 182

First, we distribute 4 to the terms inside the parentheses:

4x + 20 - 5 = 8x + 182

Next, we combine like terms:

4x + 15 = 8x + 182

To isolate the variable, we can subtract 4x from both sides:

15 = 8x - 4x + 182

This simplifies to:

15 = 4x + 182

Next, we can subtract 182 from both sides:

-167 = 4x

Finally, we divide both sides by 4 to solve for x:

-167/4 = x

Therefore, the solution for the given equation is x = -167/4.

The equation has one solution.

Answer 8

how many solutions 4(x + 5) - 5 = 8x+18/2

Answer 9

[AI-generated answer — Bot GPT 3.5]

To determine the number of solutions for the given equation, we can simplify and solve it:

4(x + 5) - 5 = 8x + 18/2

First, we simplify the equation by performing the division:

4(x + 5) - 5 = 8x + 9

Next, we distribute 4 to the terms inside the parentheses:

4x + 20 - 5 = 8x + 9

Next, we combine like terms:

4x + 15 = 8x + 9

To isolate the variable, we can subtract 8x from both sides:

-4x + 15 = 9

Next, we subtract 15 from both sides:

-4x = -6

Finally, dividing both sides by -4 to solve for x:

x = -6 / -4

Simplifying the division:

x = 3/2 or x = 1.5

Therefore, the equation has one solution, which is x = 3/2 or x = 1.5.

Answer 10

It is given that 2(3 + x) = 6 + 2x. This is an example of the ___________ property.

Responses
A associativeassociative
B commutativecommutative
C distributivedistributive
D identity

Answer 11

[AI-generated answer — Bot GPT 3.5]

To determine the property demonstrated in the given equation 2(3 + x) = 6 + 2x, we can observe the equation and its structure.

The equation follows the distributive property, which states that multiplying a number by the sum of two other numbers is the same as multiplying each term separately and then adding the results.

In this case, we are multiplying 2 by the sum of 3 and x. Applying the distributive property, we have:

2(3 + x) = 6 + 2x

Therefore, the correct answer is C) distributive.