SOLUTION: Hello, I need help with the following math problem: How many pairs of integers (b,c) satisfy the equation {{{ (b+7)/(b+4)=c/9 }}}? Thank you, Nicole

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Hello, I need help with the following math problem: How many pairs of integers (b,c) satisfy the equation {{{ (b+7)/(b+4)=c/9 }}}? Thank you, Nicole      Log On


   

Question 1046468: Hello,
I need help with the following math problem:
How many pairs of integers (b,c) satisfy the equation +%28b%2B7%29%2F%28b%2B4%29=c%2F9+?
Thank you,
Nicole
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

The equation %28b%2B7%29%2F%28b%2B4%29=c%2F9
is equivalent to 1%2B3%2F%28b%2B4%29+=+c%2F9
-> 3%2F%28b%2B4%29+=+c%2F9+-+1, after transposing 1 to the right side;
-> 3%2F%28b%2B4%29+=+%28c-9%29%2F9, after combining terms,
-> 27+=+%28b%2B4%29%28c-9%29, after cross multiplying.
Since both b%2B4 and c+-+9 are supposed to be integers, there are eight possibilities:
1) b%2B4+=+1, c-9+=+27 -> highlight%28b+=+-3%29, highlight%28c=36%29
2)+b%2B4+=+3, c-9+=+9 -> highlight%28b+=+-1%29, highlight%28c=18%29
3) b%2B4+=+9, c-9+=+3 -> highlight%28b+=+5%29, highlight%28c=12%29
4) b%2B4+=+27, c-9+=+1 -> highlight%28b+=+23%29, highlight%28c=10%29
5) b%2B4+=+-1, c-9+=+-27 -> highlight%28b+=+-5%29, highlight%28c=-18%29
6) b%2B4+=+-3, c-9+=+-9 -> highlight%28b+=+-7%29, highlight%28c=0%29
7) b%2B4+=+-9, c-9+=+-3 -> highlight%28b+=+-13%29, highlight%28c=6%29
8) b%2B4+=+-27, c-9+=+-1 ->highlight%28b+=+-31%29, highlight%28c=8%29
Therefore there are 8 pairs of integers that will satisfy the original equation.