Question

Multiple Representations The diameter of a circle is 14 in. 4 x plus 3 represents the radius. Write two equations that you can use to find the value of x. Then find the value of x.
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Part 1
Which equations can you use to solve for​ x? Select two equations.
A.2 left parenthesis 4 x plus 3 right parenthesis equals 7
2 left parenthesis 4 x plus 3 right parenthesis equals 7
B.4 x plus 3 equals 7
4 x plus 3 equals 7
C.4 x plus 3 equals 14
4 x plus 3 equals 14
D.2 left parenthesis 4 x plus 3 right parenthesis equals 14
2 left parenthesis 4 x plus 3 right parenthesis equals 14

Answer 1

To find the value of \( x \) based on the information given:

1. We know that the diameter of the circle is 14 inches, which means the radius is half of the diameter. Therefore, the radius is:
\[
\text{Radius} = \frac{\text{Diameter}}{2} = \frac{14}{2} = 7 \text{ in.}
\]

2. You've mentioned that \( 4x + 3 \) represents the radius. Since we already established that the radius is 7 inches, we can set up an equation:
\[
4x + 3 = 7.
\]

Now let's look at the provided options:

- **A: \( 2(4x + 3) = 7 \)** — This equation does not represent the radius correctly.
- **B: \( 4x + 3 = 7 \)** — This is a correct representation of the radius.
- **C: \( 4x + 3 = 14 \)** — This represents the diameter, not the radius, so it is incorrect for this situation since we need the radius.
- **D: \( 2(4x + 3) = 14 \)** — This equation effectively represents the diameter but is complex since we are looking directly at the radius.

The two equations that could be used to solve for \( x \):

1. \( 4x + 3 = 7 \) (Option B)
2. \( 2(4x + 3) = 14 \) (Option D)

### Solving for x using the correct equation:
Using **Option B**:
\[
4x + 3 = 7
\]
Subtract 3 from both sides:
\[
4x = 7 - 3
\]
\[
4x = 4
\]
Now divide by 4:
\[
x = 1.
\]

### Finally:
- The value of \( x \) is **1**.