Question

How many solutions will the equation 4h+6=14−12(16−8h)
have?(1 point)
Responses

infinite/many solutions
infinite/many solutions

one solution
one solution

no solution
no solution

two solutions

Answer 1

it's wrong

Answer 2

To determine how many solutions the equation \( 4h + 6 = 14 - 12(16 - 8h) \) has, let's first simplify the equation step by step.

1. Start by distributing \( -12 \) on the right side:
\[
4h + 6 = 14 - 12 \cdot 16 + 12 \cdot 8h
\]
\[
4h + 6 = 14 - 192 + 96h
\]
\[
4h + 6 = -178 + 96h
\]

2. Next, let's move all terms involving \( h \) to one side and constant terms to the other:
Subtract \( 4h \) from both sides:
\[
6 = -178 + 96h - 4h
\]
\[
6 = -178 + 92h
\]

3. Now, let's isolate \( h \):
Add \( 178 \) to both sides:
\[
6 + 178 = 92h
\]
\[
184 = 92h
\]
Divide both sides by \( 92 \):
\[
h = \frac{184}{92} = 2
\]

Since we found one specific value for \( h \), the equation has **one solution**.

Thus, the answer is: **one solution**.

Answer 3

Let's re-evaluate the equation step-by-step to find any mistakes in my previous calculations and ensure we reach the correct conclusion.

Given the equation:
\[
4h + 6 = 14 - 12(16 - 8h)
\]

1. **Distribute the -12 on the right-hand side:**
\[
4h + 6 = 14 - 12 \cdot 16 + 12 \cdot 8h
\]
\[
4h + 6 = 14 - 192 + 96h
\]
\[
4h + 6 = -178 + 96h
\]

2. **Rearranging the equation:**
Move the \( 4h \) to the right side and the constant \( -178 \) to the left:
\[
4h - 96h = -178 - 6
\]
\[
-92h = -184
\]

3. **Solving for \( h \):**
Divide both sides by -92:
\[
h = \frac{-184}{-92} = 2
\]

Thus, we have determined that \( h = 2 \), implying there is one specific solution to the equation.

Here's the conclusion:

The original equation \( 4h + 6 = 14 - 12(16 - 8h) \) has **one solution**.

If you believe I still made an error or if there are additional details about the question that I should consider, please let me know!