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# Find all solutions to:

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116
5

Find all solutions to:

ab + 2a + 2b = −16

ac + 2a + 2c = 26

bc + 2b + 2c = −14.

Feb 22, 2021

### 5+0 Answers

#1
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Find all solutions to:

ab + 2a + 2b = −16

ac + 2a + 2c = 26

bc + 2b + 2c = −14

Hello juananother!

$$ab + 2a + 2b = −16\\ a(b+2)+2b=-16\\ a= -\dfrac{16+2b}{b+2}\\ \ \\ ac + 2a + 2c = 26\\ a(c+2)+2c=26\\ a= \dfrac{26-2c}{c+2}$$

$$-\dfrac{16+2b}{b+2}=\dfrac{26-2{ \color{BrickRed} c}}{{ \color{BrickRed} c}+2}$$

$$bc + 2b + 2c = −14\\ \color{BrickRed} c=( -\dfrac{14+2b}{b+2})$$

$$-\dfrac{16+2b}{b+2}= \dfrac{26-2{ \color{BrickRed} (-\dfrac{14+2b)}{b+2}}}{{ \color{BrickRed} (-\dfrac{14+2b}{b+2})}+2}$$

$$b= -\dfrac{8}{3}$$

$$c= -\dfrac{14+2\cdot (-\dfrac{8}{3})}{-\dfrac{8}{3}+2}$$

$$c=-13$$

$$a= \dfrac{26-2(-13)}{(-13)+2}$$

$$a=-\dfrac{52}{11}$$ !

Feb 22, 2021
edited by asinus  Feb 22, 2021
edited by asinus  Feb 22, 2021
#2
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There have been at least two other (with better methods than this one) answers to this question posted within the last week.

Look for them.

Feb 22, 2021
#3
+1

"with better methods than this one" Is there any better math than right? At school? !

asinus  Feb 22, 2021
#4
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Some methods are certainly better than others.

(Both good and bad methods should lead to the correct answer.)

Have you looked at the earlier answers ?

Guest Feb 22, 2021
#5
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Hi asinus and guest,

Thanks for answering asinus.

Your original answer of a=2 was a part of a correct answer.

I substituted your other values in and I do not think that they work.

Guest is right.

textot  gave a very cool answer for this.

Take a look https://web2.0calc.com/questions/find-all-solutions-to

Melody  Feb 22, 2021