Solve 88-36t/frac{36t-24{t}}=1/2x-1/3

$$\frac{ 88-36t }{ \frac{ 36t-24 }{ t } } = \frac{ 1 }{ 2 } x- \frac{ 1 }{ 3 }$$

Solve for x

$$x=-\frac{2\left(-9t^{2}+25t-2\right)}{3\left(2-3t\right)}$$

$$t\neq \frac{2}{3}\text{ and }t\neq 0$$

Steps for Solving Linear Equation

Divide 88-36t by \frac{36t-24}{t} by multiplying 88-36t by the reciprocal of \frac{36t-24}{t}.

$$\frac{\left(88-36t\right)t}{36t-24}=\frac{1}{2}x-\frac{1}{3}$$

Factor the expressions that are not already factored in \frac{\left(88-36t\right)t}{36t-24}.

$$\frac{4t\left(-9t+22\right)}{12\left(3t-2\right)}=\frac{1}{2}x-\frac{1}{3}$$

Cancel out 4 in both numerator and denominator.

$$\frac{t\left(-9t+22\right)}{3\left(3t-2\right)}=\frac{1}{2}x-\frac{1}{3}$$

Use the distributive property to multiply t by -9t+22.

$$\frac{-9t^{2}+22t}{3\left(3t-2\right)}=\frac{1}{2}x-\frac{1}{3}$$

Use the distributive property to multiply 3 by 3t-2.

$$\frac{-9t^{2}+22t}{9t-6}=\frac{1}{2}x-\frac{1}{3}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{1}{2}x-\frac{1}{3}=\frac{-9t^{2}+22t}{9t-6}$$

Add \frac{1}{3} to both sides.

$$\frac{1}{2}x=\frac{-9t^{2}+22t}{9t-6}+\frac{1}{3}$$

Factor 9t-6.

$$\frac{1}{2}x=\frac{-9t^{2}+22t}{3\left(3t-2\right)}+\frac{1}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\left(3t-2\right) and 3 is 3\left(3t-2\right). Multiply \frac{1}{3} times \frac{3t-2}{3t-2}.

$$\frac{1}{2}x=\frac{-9t^{2}+22t}{3\left(3t-2\right)}+\frac{3t-2}{3\left(3t-2\right)}$$

Since \frac{-9t^{2}+22t}{3\left(3t-2\right)} and \frac{3t-2}{3\left(3t-2\right)} have the same denominator, add them by adding their numerators.

$$\frac{1}{2}x=\frac{-9t^{2}+22t+3t-2}{3\left(3t-2\right)}$$

Combine like terms in -9t^{2}+22t+3t-2.

$$\frac{1}{2}x=\frac{-9t^{2}+25t-2}{3\left(3t-2\right)}$$

Factor the expressions that are not already factored in \frac{-9t^{2}+25t-2}{3\left(3t-2\right)}.

$$\frac{1}{2}x=\frac{-9\left(t-\left(-\frac{1}{18}\sqrt{553}+\frac{25}{18}\right)\right)\left(t-\left(\frac{1}{18}\sqrt{553}+\frac{25}{18}\right)\right)}{3\left(3t-2\right)}$$

Cancel out 3 in both numerator and denominator.

$$\frac{1}{2}x=\frac{-3\left(t-\left(-\frac{1}{18}\sqrt{553}+\frac{25}{18}\right)\right)\left(t-\left(\frac{1}{18}\sqrt{553}+\frac{25}{18}\right)\right)}{3t-2}$$

Multiply both sides of the equation by 2\left(3t-2\right), the least common multiple of 2,3t-2.

$$\frac{1}{2}x\times 2\left(3t-2\right)=2\left(-1\right)\times 3\left(t-\left(-\frac{1}{18}\sqrt{553}+\frac{25}{18}\right)\right)\left(t-\left(\frac{1}{18}\sqrt{553}+\frac{25}{18}\right)\right)$$

Reorder the terms.

$$\frac{1}{2}\times 2x\left(3t-2\right)=-2\times 3\left(t-\left(-\frac{1}{18}\sqrt{553}+\frac{25}{18}\right)\right)\left(t-\left(\frac{1}{18}\sqrt{553}+\frac{25}{18}\right)\right)$$

Cancel out 2 on both sides.

$$\frac{1}{2}x\left(3t-2\right)=-3\left(t-\left(-\frac{1}{18}\sqrt{553}+\frac{25}{18}\right)\right)\left(t-\left(\frac{1}{18}\sqrt{553}+\frac{25}{18}\right)\right)$$

Use the distributive property to multiply \frac{1}{2}x by 3t-2.

$$\frac{3}{2}tx-x=-3\left(t-\left(-\frac{1}{18}\sqrt{553}+\frac{25}{18}\right)\right)\left(t-\left(\frac{1}{18}\sqrt{553}+\frac{25}{18}\right)\right)$$

To find the opposite of -\frac{1}{18}\sqrt{553}+\frac{25}{18}, find the opposite of each term.

$$\frac{3}{2}tx-x=-3\left(t+\frac{1}{18}\sqrt{553}-\frac{25}{18}\right)\left(t-\left(\frac{1}{18}\sqrt{553}+\frac{25}{18}\right)\right)$$

To find the opposite of \frac{1}{18}\sqrt{553}+\frac{25}{18}, find the opposite of each term.

$$\frac{3}{2}tx-x=-3\left(t+\frac{1}{18}\sqrt{553}-\frac{25}{18}\right)\left(t-\frac{1}{18}\sqrt{553}-\frac{25}{18}\right)$$

Use the distributive property to multiply -3 by t+\frac{1}{18}\sqrt{553}-\frac{25}{18}.

$$\frac{3}{2}tx-x=\left(-3t-\frac{1}{6}\sqrt{553}+\frac{25}{6}\right)\left(t-\frac{1}{18}\sqrt{553}-\frac{25}{18}\right)$$

Use the distributive property to multiply -3t-\frac{1}{6}\sqrt{553}+\frac{25}{6} by t-\frac{1}{18}\sqrt{553}-\frac{25}{18} and combine like terms.

$$\frac{3}{2}tx-x=-3t^{2}+\frac{25}{3}t+\frac{1}{108}\left(\sqrt{553}\right)^{2}-\frac{625}{108}$$

The square of \sqrt{553} is 553.

$$\frac{3}{2}tx-x=-3t^{2}+\frac{25}{3}t+\frac{1}{108}\times 553-\frac{625}{108}$$

Multiply \frac{1}{108} and 553 to get \frac{553}{108}.

$$\frac{3}{2}tx-x=-3t^{2}+\frac{25}{3}t+\frac{553}{108}-\frac{625}{108}$$

Subtract \frac{625}{108} from \frac{553}{108} to get -\frac{2}{3}.

$$\frac{3}{2}tx-x=-3t^{2}+\frac{25}{3}t-\frac{2}{3}$$

Combine all terms containing x.

$$\left(\frac{3}{2}t-1\right)x=-3t^{2}+\frac{25}{3}t-\frac{2}{3}$$

The equation is in standard form.

$$\left(\frac{3t}{2}-1\right)x=-3t^{2}+\frac{25t}{3}-\frac{2}{3}$$

Divide both sides by \frac{3}{2}t-1.

$$\frac{\left(\frac{3t}{2}-1\right)x}{\frac{3t}{2}-1}=\frac{-3t^{2}+\frac{25t}{3}-\frac{2}{3}}{\frac{3t}{2}-1}$$

Dividing by \frac{3}{2}t-1 undoes the multiplication by \frac{3}{2}t-1.

$$x=\frac{-3t^{2}+\frac{25t}{3}-\frac{2}{3}}{\frac{3t}{2}-1}$$

Divide -3t^{2}+\frac{25t}{3}-\frac{2}{3} by \frac{3}{2}t-1.

$$x=\frac{2\left(-9t^{2}+25t-2\right)}{3\left(3t-2\right)}$$

Solve for t

$$\left\{\begin{matrix}\\t=\frac{\sqrt{81x^{2}-468x+2212}}{36}-\frac{x}{4}+\frac{25}{18}\text{, }&\text{unconditionally}\\t=-\frac{\sqrt{81x^{2}-468x+2212}}{36}-\frac{x}{4}+\frac{25}{18}\text{, }&x\neq \frac{2}{3}\end{matrix}\right.$$