help!!!!!! super confused

Question

let a and b be real numbers. the solutions of the equation ax + b = 30/x+2  - 1 are 8 and -8. What are the values of a and b? Explain your answer.

Answer 1

ax+b=30/(x+2)-1

multiply both sides by x+2

ax(x+2)+b(x+2)=(x+2)(30/[x+2])-(x+2)

ax^2+2ax+bx+2b=30-(x+2)

ax^2+2ax+bx+2b=30-x-2

get all terms on one side

ax^2+2ax+bx+2b-30+x+2=0

combine similar terms and get a coefficient for x^2 and for x and determine the constant

ax^2+(2a+b+1)x+(2b-28)=0

replace x by 8

64a+16a+8b+8+2b-28=0

combine similar terms

80a+10b-20=0

divide all terms by 10

8a+b-2=0

replace x by -8 in the same equation as before

64a-16a-8b-8+2b-28=0

48a-6b-36=0

divide all terms by 6

8a-b-6=0

you now have a system of linear equations

8a+b-2=0

8a-b-6=0

add these equations together

16a-8=0

16a=8

a=8/16

a=1/2

replace a by 1/2 to determine b (use either equation)

8(1/2)-b-6=0

4-b-6=0

-b-2=0

b=-2

a=1/2 and b=-2

replace a and b by these values in the original equation with x=8 and then with x=-8 and you will see that you get a true statement both times

 

 

Answer 2

first get two equations: the first by substituting 8 for x and the second by substituting -8 for x

then solve the two equation for a and b , you will get a=1/2 and b= -2

Answer 3

ax+b = 30/(x+2) - 1

 

Since 8 is a solution, 8a+b = 30/10 - 1.  So, 8a+b = 2

 

Since -8 is a solution, 8a+b = 30/(-6) -1, So, -8a+b=-6

 

Adding the equations, we have 2b = -4.  So, b = -2

 

Since 8a+b = 2 and b = -2, we get 8a - 2 = 2

 

                                                            8a = 4

 

                                                              a = 1/2

 

Answer 4

I think it is easier to substitute for x initially. The explanation of the answer lies in the fact that -8 and 8 are the values of x that will make the equation true for particular values of a and b. An equation is a statement of conditional equality. Substituting for x gives 8a+b=30/10-1=2 -8a+b=30/-6 -1=-5-1=-6 Add the two equations to eliminate a 2b=2+(-6)=-4 b=-2 Then substitute for b in one of the equations 8a+(-2)=2 8a=4 a=1/2