Question

Determine what kind of solution the following equations have and why.(2 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
1) 3x+12+x=8+4+x
has Response area

2) 4(x+1)=4x+1
has Response area

Answer 1

To determine the kind of solution each equation has, let’s analyze them one by one.

1) **Equation: \(3x + 12 + x = 8 + 4 + x\)**

Simplifying the left side:
\[
3x + x + 12 = 4x + 12
\]

Simplifying the right side:
\[
8 + 4 = 12
\]
So the equation becomes:
\[
4x + 12 = x + 12
\]

Now, isolate \(x\):
Subtract \(x + 12\) from both sides:
\[
4x + 12 - x - 12 = 0 \implies 3x = 0 \implies x = 0
\]

**Conclusion:** This equation has a **unique solution**: \(x = 0\).

2) **Equation: \(4(x + 1) = 4x + 1\)**

Expand the left side:
\[
4x + 4 = 4x + 1
\]

Now, isolate \(x\):
Subtract \(4x\) from both sides:
\[
4 = 1
\]
This statement is false.

**Conclusion:** This equation has **no solution**, as the two sides cannot be equal.

Putting it all together:

1) **Response for** \(3x + 12 + x = 8 + 4 + x\): **unique solution**

2) **Response for** \(4(x + 1) = 4x + 1\): **no solution**