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For this specific problem, I somehow keep coming up with the wrong answer. Can someone help me?

For the problem, I need to solve for $g$.

$$12g = 12 \left(\frac{2}{3g} - 1\right) + 11$$

Here is how I am trying to solve it:

$$12 \cdot \frac{2}{3g} = 8g$$

$$12 \cdot -1 = -12$$

$$12g = 8g - 12 + 11$$

(Then I subtract $8g$ from $12g$ and $8g$)

$4g = -12 + 11$

(Then I add $12$ to $-12$ and $11$)

$$4g = 23$$

(Then I divide $4$ from $4g$ and $23$)

$$g = 5.75$$

So, shouldn't $g$ equal $5.75$? But when I plug it in the equation to check my answer, both answers do not match. Where am I going wrong?

Blue
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Janaia Hunt
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    Is this $$12g-11=12\cdot \frac{2}{3g-1}$$? – Dr. Sonnhard Graubner Sep 26 '18 at 17:10
  • $12g=12(\dfrac{2}{3g})-12+11\iff12g=8g-1\iff 4g=-1$ – Piquito Sep 26 '18 at 17:17
  • No, $-12+11=-1$. –  Sep 26 '18 at 17:39
  • Caution, the notation $2/3g$ is ambiguous ($\dfrac{2}3g$ or $\dfrac2{3g}$) and deceived several answerers. –  Sep 26 '18 at 17:43
  • @Piquito You forgot to multiply both sides by $g$ to get rid of the denominator. Instead, you just move $8g$ to the numberator. – poetasis Sep 26 '18 at 17:46
  • Thanks everyone for your help! I am sorry about the confusion, but the problem is using 2/3 as a whole fraction multiplied by g, not only the 3 being multiplied by g. – Janaia Hunt Sep 27 '18 at 00:10
  • @Janaia Hunt please do mark an answer as correct by checking the check mark on the left side of the post. – Max0815 Sep 27 '18 at 04:12
  • @Janaia Hunt please do mark an answer as correct by checking the check mark on the left side of the post. – Max0815 Sep 27 '18 at 21:05
  • @Janaia Hunt please do mark an answer as correct by checking the check mark on the left side of the post. peotasis has already reminded you too. – Max0815 Sep 28 '18 at 00:34
  • @Janaia Hunt please do mark an answer as correct by checking the check mark on the left side of the post. peotasis has already reminded you too. – Max0815 Sep 28 '18 at 22:44
  • @Janaia Hunt please do mark an answer as correct by checking the check mark on the left side of the post. – Max0815 Feb 08 '19 at 19:28
  • @Janaia Hunt please do mark an answer as correct by checking the check mark on the left side of the post. – Max0815 Feb 17 '19 at 16:40

2 Answers2

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So you have the equation $12g = 12 (\frac{2}{3g} - 1) + 11$. You want to find $g$. $$12g=\frac{24}{3g}-12+11$$ $$\implies 12g=\frac{8}{g}-1$$ $$\implies 12g^2=8-g$$ $$12g^2+g-8=0$$ $$g=\boxed{\frac{-1\pm\sqrt{385}}{24}}$$

However, if you mean $12g=12(\frac{2}{3g-1})+11$, then we have $$12g=\frac{24}{3g-1}+11$$ $$\implies 12g(3g-1)=24+33g-11$$ $$\implies 36g^2-12g=33g+13$$ $$\implies 36g^2-45g-13=0$$ This is a quadratic, so use the quadratic formula getting $$g=\boxed{\frac{15\pm\sqrt{433}}{24}}$$ I’m sure there is some limit of $g$ to eliminate on of the solutions, such as “no negative numbers”, so I think you can take it from there... given that I’m right :P.

Max0815
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  • @Janaia Hunt It seems that Blue edited your question 19hrs ago, as I look. If that is the correct form of the equation, I believe that Max0815 and I both gave the correct answer except that I omitted the quadratic equation part that he included. Mine was down voted but, if either of them is correct, please check it as such so that one of us can get the bonus point. Let us know if there is still more to the mystery and thanks. – poetasis Sep 27 '18 at 16:39
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You can convert it to a quadratic equation, where it can be solved rather simply.
$$12g = 12(\frac{2}{3g} -1)+11$$ $$12g = \frac{24}{3g}-12+11 \implies 12g = \frac{8}{g}-1$$ $$\implies 12g^2 = 8-g\text{ (Multiply all terms by }g\text{.)}$$ $$\implies 12g^2+g-8 = 0$$
There is no simple way to factor this as we don’t have any integers which add up to $b$ and multiple to get $ac$. So, just use the quadratic formula.
$$\implies x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\text{ for }a = 12, b = 1, c = -8$$
$$\implies x = \frac{-1\pm\sqrt{1^2-4(12)(-8)}}{2(12)}$$
$$\implies x = \frac{-1\pm\sqrt{385}}{24}$$

Max0815
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KM101
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