Question

Consider solving the equation for x.

3(3x + 4) - 5 = ax + b
Which statements are true about the solution to the equation when substituting values for a and b as specified?
Select all that apply.

A. If a=−3 and b=4 , then there is exactly one solution to the equation.
B. If a=−9 and b=13 , then there is no solution to the equation.
C. If a=9 and b=−2 , then there is no solution to the equation.
D. If a=−3 and b=−5 , then there are infinitely many solutions to the equation.
E. If a=9 and b=7 , then there is exactly one solution to the equation.

Answer 1

So let's simplify the equation first.

9x + 12 - 5 = ax + b
9x + 7 = ax + b
9x - ax - b = 7

Alright now plug in the numbers for each option.
I don't feel like doing all of them but I did A (one solution) and B (no solution because the two 9x's cancel out).

Answer 2

A. If a=−3 and b=4 , then there is exactly one solution to the equation.

C. If a=9 and b=−2 , then there is no solution to the equation.

Explanation:

The given equation is

`3(3x+4)-5=ax+b`

To solve for
`x`, we need to use the distributive property

`3(3x+4)-5=ax+b\\9x+12-5=ax+b\\9x-ax=b-7\\x(9-a)=b-7`

To find the right statement, we need to the given values.

For a=-3 and b=4,

`x(9-a)=b-7\\x(9-(-3))=4-7\\x(9+3)=-3\\12x=-3\\x=(-3)/(12) =-(1)/(4)`

So, choice A is true, because there's one solution to the equation.

For a=-9 and b=13,

`x(9-a)=b-7\\x(9-(-9))=13-7\\18x=6\\x=(6)/(18)=(1)/(3)`

The choice B is false.

For a=9 and b=-2

`x(9-a)=b-7\\x(9-9)=-2-7\\0x=-9\\0=-9`

Choice C is correct, because there's no solution, because zero is not equal to -9.

For a=-3 and b=-5

`x(9-a)=b-7\\x(9-(-3))=-5-7\\12x=-12\\x=(-12)/(12)=-1`

Choice D is false, becuase there's only one solution.

For a=9 and b=7

`x(9-a)=b-7\\x(9-9)=7-7\\0x=0\\0=0`

Choice E is false, because there are infinite solutions in ths case.

Therefore, the right choices are A and C.