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Find all solutions to

ab+2a+2b=-16

ac+2a+2c=26

bc+2b+2c=-14

 Feb 18, 2021
 #1
avatar+112966 
+2

My algebra was very long-winded and I have not finished it but

 

I used the first one to get a in terms of b       a(b)

then the last one to get b in terms of c          b(c)

then i substituted the b(c)  into the a to get a(b)  in terms of a(c)

 

Then I substituted this into the middle one and solved for c

etc

 

There is probably a better way.   

 Feb 18, 2021
edited by Melody  Feb 18, 2021
 #2
avatar+299 
+2

Use Simon's favorite factoring trick; it's going to make it so much easier:

\((a+2)(b+2)=-12\\ (a+2)(c+2)=30\\ (b+2)(c+2)=-10\)

Multiply all the equations to get:

\(((a+2)(b+2)(c+2))^2=3600\\ (a+2)(b+2)(c+2)= \pm 60\)

In the first case where the right-hand side of the equation above is positive, to solve for a, divide \((a+2)(b+2)(c+2)=60\) by \((b+2)(c+2)=-10\) to get \(a+2=-6 \rightarrow a=-8\). Do that the same with b and c to get \(b = 0\) and \(c=-7\).

In the second case where the right-hand side of the equation is negative, do the exact same thing mentioned above to get \(a = 4, b = -4, c = 3\).

Therefore, the solutions to this system of the equation are:

\(\boxed{a = -8, b = 0, c = -7 \text{ or } a = 4, b = -4, c = 3} \)

 Feb 20, 2021
 #3
avatar+112966 
+1

That is really cool Textot.

Thanks very much for showing me.   cool

 Feb 20, 2021

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