How do you solve #\frac { 4a + 5} { a + 8} = \frac { 4a + 6} { a + 9}#?

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Answer 1 · 2018-06-20T15:32:03

#a=1#

First of all, we must assume that #a \ne -8# and #a \ne -9#, otherwise one of the two denominators would vanish.

With this assumption, we can multiply both sides by #(a+8)(a+9)# to get

#cancel((a+8))(a+9)\frac{4a+5}{cancel(a+8)} = \frac{4a+6}{cancel(a+9)}(a+8)cancel((a+9))#

So, the equation is

#(a+9)(4a+5) = (4a+6)(a+8)#

Expand both sides to get

#cancel(4 a^2) + 41 a + 45 = cancel(4 a^2) + 38 a + 48#

Subtract #38a# from both sides:

#3a +45 = 48#

Subtract #45# from both sides:

#3a =3#

Divide both sides by #3#:

#a=1#

The solution respects the conditions #a \ne -8# and #a\ne -9#, so we can accept it.