Question

Which equations have exactly one solution? (choose 2)




Responses
A 6m – 9 = 6m6 m – 9 = 6 m
B 24a – 22 = -4(1 – 6a)



24 a – 22 = -4(1 – 6 a )
C 5n + 34 = -2(1 – 7n)
5 n + 34 = -2(1 – 7 n )
D 3n – 5 = -8(6 + 5n)
3 n – 5 = -8(6 + 5 n )
E p – 6 = -8 + pp – 6 = -8 + p

Answer 1

To determine which equations have exactly one solution, we can simplify and solve each of the given equations.

### Option A:
\[
6m - 9 = 6m
\]

Subtract \(6m\) from both sides:
\[
6m - 6m - 9 = 0 \implies -9 = 0
\]
This is a contradiction and means there are no solutions.

### Option B:
\[
24a - 22 = -4(1 - 6a)
\]

Distributing on the right side:
\[
24a - 22 = -4 + 24a
\]

Subtracting \(24a\) from both sides:
\[
-22 = -4
\]
This is also a contradiction and means there are no solutions.

### Option C:
\[
5n + 34 = -2(1 - 7n)
\]

Distributing on the right side:
\[
5n + 34 = -2 + 14n
\]

Rearranging to isolate \(n\):
\[
5n - 14n + 34 + 2 = 0 \implies -9n + 36 = 0
\]
Solving for \(n\):
\[
-9n = -36 \implies n = 4
\]
This equation has exactly one solution: \(n = 4\).

### Option D:
\[
3n - 5 = -8(6 + 5n)
\]

Distributing on the right side:
\[
3n - 5 = -48 - 40n
\]

Rearranging gives:
\[
3n + 40n = -48 + 5 \implies 43n = -43
\]
So:
\[
n = -1
\]
This equation also has exactly one solution: \(n = -1\).

### Option E:
\[
p - 6 = -8 + p
\]

Subtracting \(p\) from both sides gives:
\[
-6 = -8
\]
This is a contradiction, meaning there are no solutions.

### Summary:
The equations that have exactly one solution are:
- **Option C: \(5n + 34 = -2(1 - 7n)\)**
- **Option D: \(3n - 5 = -8(6 + 5n)\)**