Question

How many solutions will the equation 4h+6=14−1/2(16−8h)
have?(1 point)/./
Responses

two solutions
two solutions - no response given

one solution
one solution - incorrect

infinite/many solutions
infinite/many solutions - no response given

no solution

Answer 1

Using properties of equalities, find the solution to the equation 3/7x+6/7−2x=4
.(1 point)
Responses

22
22

−2
negative 2

2
2

−22

Answer 2

Identify the number of equations that have the same solution set.

Equation A: 45(3f−7)=25(4f−8)


Equation B: 59g−2=79


Equation C: −2(4−2h)=−3(1−h)


(1 point)
Responses

none
none

2
2

1
1

3

Answer 3

To determine how many solutions the equation \( 4h + 6 = 14 - \frac{1}{2}(16 - 8h) \) has, we first simplify the equation step-by-step.

1. Start by simplifying the right side of the equation:
\[
-\frac{1}{2}(16 - 8h) = -\frac{1}{2} \cdot 16 + \frac{1}{2} \cdot 8h = -8 + 4h
\]

Therefore, the equation becomes:
\[
4h + 6 = 14 - 8 + 4h
\]

2. This simplifies to:
\[
4h + 6 = 6 + 4h
\]

3. Now, subtract \( 4h \) from both sides:
\[
6 = 6
\]

Since the left side equals the right side and does not depend on \( h \), this indicates that the equation is true for any value of \( h \).

Thus, the equation has **infinite/many solutions**.

The correct answer is: **infinite/many solutions**.

Answer 4

To solve the equation \(\frac{3}{7}x + \frac{6}{7} - 2x = 4\), we will follow the steps below:

1. **Combine like terms**: First, rewrite \(2x\) in terms of fractions so it has a common denominator with \(\frac{3}{7}x\):
\[
-2x = -\frac{14}{7}x
\]
Now we can rewrite the equation:
\[
\frac{3}{7}x + \frac{6}{7} - \frac{14}{7}x = 4
\]

2. **Combine the \(x\) terms**:
\[
\left(\frac{3}{7}x - \frac{14}{7}x\right) + \frac{6}{7} = 4
\]
This simplifies to:
\[
-\frac{11}{7}x + \frac{6}{7} = 4
\]

3. **Isolate the \(x\) term**: First, subtract \(\frac{6}{7}\) from both sides:
\[
-\frac{11}{7}x = 4 - \frac{6}{7}
\]

To subtract, we can convert \(4\) to a fraction with a denominator of \(7\):
\[
4 = \frac{28}{7}
\]
So,
\[
4 - \frac{6}{7} = \frac{28}{7} - \frac{6}{7} = \frac{22}{7}
\]

Now we have:
\[
-\frac{11}{7}x = \frac{22}{7}
\]

4. **Solve for \(x\)**: Multiply both sides by \(-\frac{7}{11}\) to isolate \(x\):
\[
x = \frac{22}{7} \cdot -\frac{7}{11} = -\frac{22}{11} = -2
\]

Thus, the solution to the equation is \( x = -2 \).

The correct answer is: **−2**.

Answer 5

To identify the number of equations that have the same solution set, we will solve each equation one by one.

### Equation A: \( 45(3f - 7) = 25(4f - 8) \)

1. Distribute both sides:
\[
135f - 315 = 100f - 200
\]

2. Rearrange to isolate \(f\):
\[
135f - 100f = 315 - 200
\]
\[
35f = 115
\]
\[
f = \frac{115}{35} = \frac{23}{7}
\]

### Equation B: \( 59g - 2 = 79 \)

1. Add 2 to both sides:
\[
59g = 81
\]
2. Divide by 59:
\[
g = \frac{81}{59}
\]

### Equation C: \( -2(4 - 2h) = -3(1 - h) \)

1. Distribute both sides:
\[
-8 + 4h = -3 + 3h
\]

2. Rearrange to isolate \(h\):
\[
4h - 3h = -3 + 8
\]
\[
h = 5
\]

### Summary of Solutions:

- **Equation A**: \( f = \frac{23}{7} \)
- **Equation B**: \( g = \frac{81}{59} \)
- **Equation C**: \( h = 5 \)

### Conclusion:
The three equations have different solutions:

1. Equation A: \( f = \frac{23}{7} \)
2. Equation B: \( g = \frac{81}{59} \)
3. Equation C: \( h = 5 \)

Since none of the equations has the same solution set, the correct answer is **none**.