Solve -4a/3a-16x=3a-16/4ax+4

$$- \frac{ 4a }{ 3a-16 } x = \frac{ 3a-16 }{ 4a } x+4$$

Solve for x (complex solution)

$$x=-\frac{16a\left(16-3a\right)}{-25a^{2}+96a-256}$$

$$a\neq \frac{48}{25}-\frac{64}{25}i\text{ and }a\neq \frac{48}{25}+\frac{64}{25}i\text{ and }a\neq \frac{16}{3}\text{ and }a\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 4a\left(3a-16\right), the least common multiple of 3a-16,4a.

$$\left(-\frac{4a}{3a-16}\right)x\times 4a\left(3a-16\right)=\left(3a-16\right)\left(3a-16\right)x+4a\left(3a-16\right)\times 4$$

Multiply 3a-16 and 3a-16 to get \left(3a-16\right)^{2}.

$$\left(-\frac{4a}{3a-16}\right)x\times 4a\left(3a-16\right)=\left(3a-16\right)^{2}x+4a\left(3a-16\right)\times 4$$

Express \left(-\frac{4a}{3a-16}\right)x as a single fraction.

$$\frac{-4ax}{3a-16}\times 4a\left(3a-16\right)=\left(3a-16\right)^{2}x+4a\left(3a-16\right)\times 4$$

Express \frac{-4ax}{3a-16}\times 4 as a single fraction.

$$\frac{-4ax\times 4}{3a-16}a\left(3a-16\right)=\left(3a-16\right)^{2}x+4a\left(3a-16\right)\times 4$$

Express \frac{-4ax\times 4}{3a-16}a as a single fraction.

$$\frac{-4ax\times 4a}{3a-16}\left(3a-16\right)=\left(3a-16\right)^{2}x+4a\left(3a-16\right)\times 4$$

Express \frac{-4ax\times 4a}{3a-16}\left(3a-16\right) as a single fraction.

$$\frac{-4ax\times 4a\left(3a-16\right)}{3a-16}=\left(3a-16\right)^{2}x+4a\left(3a-16\right)\times 4$$

Cancel out 3a-16 in both numerator and denominator.

$$-4\times 4aax=\left(3a-16\right)^{2}x+4a\left(3a-16\right)\times 4$$

Multiply a and a to get a^{2}.

$$-4\times 4a^{2}x=\left(3a-16\right)^{2}x+4a\left(3a-16\right)\times 4$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(3a-16\right)^{2}.

$$-4\times 4a^{2}x=\left(9a^{2}-96a+256\right)x+4a\left(3a-16\right)\times 4$$

Use the distributive property to multiply 9a^{2}-96a+256 by x.

$$-4\times 4a^{2}x=9a^{2}x-96ax+256x+4a\left(3a-16\right)\times 4$$

Multiply 4 and 4 to get 16.

$$-4\times 4a^{2}x=9a^{2}x-96ax+256x+16a\left(3a-16\right)$$

Use the distributive property to multiply 16a by 3a-16.

$$-4\times 4a^{2}x=9a^{2}x-96ax+256x+48a^{2}-256a$$

Multiply -4 and 4 to get -16.

$$-16a^{2}x=9a^{2}x-96ax+256x+48a^{2}-256a$$

Subtract 9a^{2}x from both sides.

$$-16a^{2}x-9a^{2}x=-96ax+256x+48a^{2}-256a$$

Combine -16a^{2}x and -9a^{2}x to get -25a^{2}x.

$$-25a^{2}x=-96ax+256x+48a^{2}-256a$$

Add 96ax to both sides.

$$-25a^{2}x+96ax=256x+48a^{2}-256a$$

Subtract 256x from both sides.

$$-25a^{2}x+96ax-256x=48a^{2}-256a$$

Combine all terms containing x.

$$\left(-25a^{2}+96a-256\right)x=48a^{2}-256a$$

Divide both sides by -25a^{2}+96a-256.

$$\frac{\left(-25a^{2}+96a-256\right)x}{-25a^{2}+96a-256}=\frac{16a\left(3a-16\right)}{-25a^{2}+96a-256}$$

Dividing by -25a^{2}+96a-256 undoes the multiplication by -25a^{2}+96a-256.

$$x=\frac{16a\left(3a-16\right)}{-25a^{2}+96a-256}$$

Solve for x

$$x=-\frac{16a\left(16-3a\right)}{-25a^{2}+96a-256}$$

$$a\neq \frac{16}{3}\text{ and }a\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 4a\left(3a-16\right), the least common multiple of 3a-16,4a.

$$\left(-\frac{4a}{3a-16}\right)x\times 4a\left(3a-16\right)=\left(3a-16\right)\left(3a-16\right)x+4a\left(3a-16\right)\times 4$$

Multiply 3a-16 and 3a-16 to get \left(3a-16\right)^{2}.

$$\left(-\frac{4a}{3a-16}\right)x\times 4a\left(3a-16\right)=\left(3a-16\right)^{2}x+4a\left(3a-16\right)\times 4$$

Express \left(-\frac{4a}{3a-16}\right)x as a single fraction.

$$\frac{-4ax}{3a-16}\times 4a\left(3a-16\right)=\left(3a-16\right)^{2}x+4a\left(3a-16\right)\times 4$$

Express \frac{-4ax}{3a-16}\times 4 as a single fraction.

$$\frac{-4ax\times 4}{3a-16}a\left(3a-16\right)=\left(3a-16\right)^{2}x+4a\left(3a-16\right)\times 4$$

Express \frac{-4ax\times 4}{3a-16}a as a single fraction.

$$\frac{-4ax\times 4a}{3a-16}\left(3a-16\right)=\left(3a-16\right)^{2}x+4a\left(3a-16\right)\times 4$$

Express \frac{-4ax\times 4a}{3a-16}\left(3a-16\right) as a single fraction.

$$\frac{-4ax\times 4a\left(3a-16\right)}{3a-16}=\left(3a-16\right)^{2}x+4a\left(3a-16\right)\times 4$$

Cancel out 3a-16 in both numerator and denominator.

$$-4\times 4aax=\left(3a-16\right)^{2}x+4a\left(3a-16\right)\times 4$$

Multiply a and a to get a^{2}.

$$-4\times 4a^{2}x=\left(3a-16\right)^{2}x+4a\left(3a-16\right)\times 4$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(3a-16\right)^{2}.

$$-4\times 4a^{2}x=\left(9a^{2}-96a+256\right)x+4a\left(3a-16\right)\times 4$$

Use the distributive property to multiply 9a^{2}-96a+256 by x.

$$-4\times 4a^{2}x=9a^{2}x-96ax+256x+4a\left(3a-16\right)\times 4$$

Multiply 4 and 4 to get 16.

$$-4\times 4a^{2}x=9a^{2}x-96ax+256x+16a\left(3a-16\right)$$

Use the distributive property to multiply 16a by 3a-16.

$$-4\times 4a^{2}x=9a^{2}x-96ax+256x+48a^{2}-256a$$

Multiply -4 and 4 to get -16.

$$-16a^{2}x=9a^{2}x-96ax+256x+48a^{2}-256a$$

Subtract 9a^{2}x from both sides.

$$-16a^{2}x-9a^{2}x=-96ax+256x+48a^{2}-256a$$

Combine -16a^{2}x and -9a^{2}x to get -25a^{2}x.

$$-25a^{2}x=-96ax+256x+48a^{2}-256a$$

Add 96ax to both sides.

$$-25a^{2}x+96ax=256x+48a^{2}-256a$$

Subtract 256x from both sides.

$$-25a^{2}x+96ax-256x=48a^{2}-256a$$

Combine all terms containing x.

$$\left(-25a^{2}+96a-256\right)x=48a^{2}-256a$$

Divide both sides by -25a^{2}+96a-256.

$$\frac{\left(-25a^{2}+96a-256\right)x}{-25a^{2}+96a-256}=\frac{16a\left(3a-16\right)}{-25a^{2}+96a-256}$$

Dividing by -25a^{2}+96a-256 undoes the multiplication by -25a^{2}+96a-256.

$$x=\frac{16a\left(3a-16\right)}{-25a^{2}+96a-256}$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{16\left(-\sqrt{4-x^{2}}+2\right)}{-3\sqrt{4-x^{2}}+4x+6}\text{, }&x\neq 0\\a=\frac{16\left(\sqrt{4-x^{2}}+2\right)}{3\sqrt{4-x^{2}}+4x+6}\text{, }&x\neq 0\text{ and }x\neq -\frac{48}{25}\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}a=\frac{16\left(-\sqrt{4-x^{2}}+2\right)}{-3\sqrt{4-x^{2}}+4x+6}\text{, }&x\neq 0\text{ and }|x|\leq 2\\a=\frac{16\left(\sqrt{4-x^{2}}+2\right)}{3\sqrt{4-x^{2}}+4x+6}\text{, }&x\neq -\frac{48}{25}\text{ and }|x|\leq 2\text{ and }x\neq 0\end{matrix}\right.$$