Question

3.
Which equations have infinitely many solutions?
Select all that apply.
A.
2x = 3x - x
B.
3x = 3(2 + x)
C.
4x = x + 4
D.
-2x = -x - 2
E.
x - 1 = 2x - (x + 1)

Answer 1

Answer:

A. 2x = 3x - x

E. x - 1 = 2x - (x + 1)

Step-by-step explanation:

To find out which linear equations have an infinitely many solutions, we must solve for the value of x:

A. 2x = 3x - x

Subtract 2x from both sides:

2x - 2x = 3x - x - 2x

0 = 3x - 3x

0 = 0  This is a true statement, which implies that any values of x will satisfy the given equation. Therefore, this linear equation has infinitely many solutions.

B. 3x = 3(2 + x)

Distribute 3 into (2 + x):

3x = 6 + 3x

Subtract 3x from both sides:

3x - 3x = 6 + 3x - 3x

0 = 6  This is a false statement.  Therefore, there is no solution.

C. 4x = x + 4

Subtract x from both sides:

4x - x = x - x + 4

3x = 4

Divide both sides by 3:

$\frac{3x}{3} = \frac{4}{3}$

x = 4/3  This is the solution to the given equation.

D. -2x = -x - 2

Add x to both sides:

-2x + x  = -x + x  - 2

-x = -2

Divide both sides by -1:

$\frac{-1x}{-1} = \frac{-2}{-1}$

x = 2  This is the solution to the given equation.

E. x - 1 = 2x - (x + 1)

Distribute -1 into (x + 1):

x - 1 = 2x - x - 1

Add 1 to both sides:

x - 1 + 1 = 2x - x - 1 + 1

x = x

Subtract x from both sides:

x - x = x - x

0 = 0  This is a true statement, which implies that any values of x will satisfy the given equation. Therefore, this linear equation has infinitely many solutions.

Given these calculations, we can see that options A and E have infinitely many solutions.

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