Question 1046468


The equation {{{(b+7)/(b+4)=c/9}}}
is equivalent to {{{1+3/(b+4) = c/9}}}

-> {{{3/(b+4) = c/9 - 1}}}, after transposing 1 to the right side;
-> {{{3/(b+4) = (c-9)/9}}}, after combining terms,
-> {{{27 = (b+4)(c-9)}}}, after cross multiplying.
Since both {{{b+4}}} and {{{c - 9}}} are supposed to be integers, there are {{{eight}}} possibilities:

1) {{{b+4 = 1}}}, {{{c-9 = 27}}} -> {{{highlight(b = -3)}}}, {{{highlight(c=36)}}}
2){{{ b+4 = 3}}}, {{{c-9 = 9}}} -> {{{highlight(b = -1)}}}, {{{highlight(c=18)}}}
3) {{{b+4 = 9}}}, {{{c-9 = 3}}}  -> {{{highlight(b = 5)}}}, {{{highlight(c=12)}}}
4) {{{b+4 = 27}}}, {{{c-9 = 1}}} -> {{{highlight(b = 23)}}}, {{{highlight(c=10)}}}
5) {{{b+4 = -1}}}, {{{c-9 = -27}}} -> {{{highlight(b = -5)}}}, {{{highlight(c=-18)}}}
6) {{{b+4 = -3}}}, {{{c-9 = -9}}} -> {{{highlight(b = -7)}}}, {{{highlight(c=0)}}}
7) {{{b+4 = -9}}}, {{{c-9 = -3}}} -> {{{highlight(b = -13)}}}, {{{highlight(c=6)}}}
8) {{{b+4 = -27}}}, {{{c-9 = -1}}} ->{{{highlight(b = -31)}}}, {{{highlight(c=8)}}}

Therefore there are {{{8}}} pairs of integers that will satisfy the original equation.