Master Simple Equations with Step-by-Step Examples | Math Homework Help

[ImAnEngineer]

Homework Statement

a=(4b-14)/(b-5)

b= ...

2. The attempt at a solution
Attempted a lot of things, but couldn't isolate the b. Is it possible at all to do this, when there are multiple terms in the numerator. If so, how?

 

[ImAnEngineer]

Never mind, problem solved.

b= 5a-14 / a-4

 

[Architeuthis Dux]

Yes, it is possible to solve for b in this equation. Here is a step-by-step example:

1. Distribute the 4 to the terms inside the parentheses:
a = (4b-14)/(b-5)
a = (4b-20+6)/(b-5)

2. Combine like terms inside the parentheses:
a = (4b-14)/(b-5)
a = (4b-14+6)/(b-5)
a = (4b-8)/(b-5)

3. Factor out a common factor of 4 in the numerator:
a = (4b-8)/(b-5)
a = 4(b-2)/(b-5)

4. Now we can see that the denominator is the difference of two terms, so we can use the difference of squares formula to simplify it:
a = 4(b-2)/((b-5)(b+5))

5. Now we can set this equal to b and solve for b:
b = 4(b-2)/((b-5)(b+5))

6. Multiply both sides by the denominator to get rid of the fraction:
b(b-5)(b+5) = 4(b-2)

7. Expand the left side using the distributive property:
b^3-5b^2+5b^2-25b = 4b-8

8. Combine like terms on both sides:
b^3-20b+8 = 0

9. Use the rational root theorem to find possible rational solutions:
Possible solutions: ±1, ±2, ±4, ±8

10. Use synthetic division to test the possible solutions:
b=-2 is a solution, so (b+2) is a factor of the polynomial.

11. Use synthetic division again to divide the polynomial by (b+2):
b^3-20b+8 = (b+2)(b^2-2b-4)

12. Use the quadratic formula to solve for the remaining solutions:
b^2-2b-4 = 0
b = (2±√(4+4(4)))/2
b = (2±√20)/2
b = 1±√5

Therefore, the solutions to the equation are b = -2, 1+√5, and 1-√5.