Question

Which of the following equations results in no solutions?

A 3(v + 5) = 2v - 4 + v
B 3(v + 5) = 2v + 15 + v
C 3(v + 5) = 2v + 15
D 3(v + 5) = 2v - 4

Answer 1

A. This gives us 3v + 15 = 3v - 4. This is no solutions because the slope is the same and the y-intercept is different.

Answer 2

`\boxed{\sf A) \:3(v + 5) = 2v - 4 + v}`

Explanation:

`\sf A )\:3(v + 5) = 2v - 4 + v`

Combine like terms:

`\sf 3\left(v+5\right)=3v-4`

Expand:

`\sf 3v+15=3v-4`

Subtract 15 from both sides:

`\sf 3v+15-15=3v-4-15`

`\sf 3v=3v-19`

Subtract 3v from both sides:

`\sf 3v-3v=3v-19-3v`

`\sf 0=-19`

`\boxed{\sf{No\:Solution}}`

_________________

`\sf B )\:3(v + 5) = 2v + 15 + v`

Combine like terms:

`\sf 3\left(v+5\right)=3v+15`

Expand:

`\sf 3v+15=3v+15`

Subtract 15 from both sides:

`\sf 3v+15-15=3v+15-15`

`\sf 3v=3v`

Subtract 3v from both sides:

`\boxed{\sf 0=0}`

_______________________

`\sf C)\: 3(v + 5) = 2v + 15`

Expand:

`\sf 3v+15=2v+15`

Subtract 15 from both sides:

`\sf 3v=2v`

Subtract 2v from both sides:

`\sf 3v-2v=2v-2v`

`\boxed{\sf v=0}`

_______________________

`\sf D)\: 3(v + 5) = 2v - 4`

Expand:

`\sf 3v+15=2v-4`

Subtract 15 from both sides:

`\sf 3v+15-15=2v-4-15`

`\sf 3v=2v-19`

Subtract 2v from both sides:

`\sf 3v-2v=2v-19-2v`

`\boxed{\sf v=-19}`

___________________________