Solve left(3a+1/4bright)left(2a-1/2bright)-2/3bleft(3/4a-frac{1}{8}bright)-3aleft(frac{2}{3}b+2aright)=-3ab

$$\left( 3a+ \frac{ 1 }{ 4 } b \right) \left( 2a- \frac{ 1 }{ 2 } b \right) - \frac{ 2 }{ 3 } b \left( \frac{ 3 }{ 4 } a- \frac{ 1 }{ 8 } b \right) -3a \left( \frac{ 2 }{ 3 } b+2a \right) = -3ab$$

Solve for a

$$\left\{\begin{matrix}\\a=-\frac{b}{12}\text{, }&\text{unconditionally}\\a\in \mathrm{R}\text{, }&b=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 3a+\frac{1}{4}b by 2a-\frac{1}{2}b and combine like terms.

$$6a^{2}-ab-\frac{1}{8}b^{2}-\frac{2}{3}b\left(\frac{3}{4}a-\frac{1}{8}b\right)-3a\left(\frac{2}{3}b+2a\right)=-3ab$$

Add 3ab to both sides.

$$6a^{2}-ab-\frac{1}{8}b^{2}-\frac{2}{3}b\left(\frac{3}{4}a-\frac{1}{8}b\right)-3a\left(\frac{2}{3}b+2a\right)+3ab=0$$

Use the distributive property to multiply -\frac{2}{3}b by \frac{3}{4}a-\frac{1}{8}b.

$$6a^{2}-ab-\frac{1}{8}b^{2}-\frac{1}{2}ba+\frac{1}{12}b^{2}-3a\left(\frac{2}{3}b+2a\right)+3ab=0$$

Combine -ab and -\frac{1}{2}ba to get -\frac{3}{2}ab.

$$6a^{2}-\frac{3}{2}ab-\frac{1}{8}b^{2}+\frac{1}{12}b^{2}-3a\left(\frac{2}{3}b+2a\right)+3ab=0$$

Combine -\frac{1}{8}b^{2} and \frac{1}{12}b^{2} to get -\frac{1}{24}b^{2}.

$$6a^{2}-\frac{3}{2}ab-\frac{1}{24}b^{2}-3a\left(\frac{2}{3}b+2a\right)+3ab=0$$

Use the distributive property to multiply -3a by \frac{2}{3}b+2a.

$$6a^{2}-\frac{3}{2}ab-\frac{1}{24}b^{2}-2ab-6a^{2}+3ab=0$$

Combine -\frac{3}{2}ab and -2ab to get -\frac{7}{2}ab.

$$6a^{2}-\frac{7}{2}ab-\frac{1}{24}b^{2}-6a^{2}+3ab=0$$

Combine 6a^{2} and -6a^{2} to get 0.

$$-\frac{7}{2}ab-\frac{1}{24}b^{2}+3ab=0$$

Combine -\frac{7}{2}ab and 3ab to get -\frac{1}{2}ab.

$$-\frac{1}{2}ab-\frac{1}{24}b^{2}=0$$

Add \frac{1}{24}b^{2} to both sides. Anything plus zero gives itself.

$$-\frac{1}{2}ab=\frac{1}{24}b^{2}$$

The equation is in standard form.

$$\left(-\frac{b}{2}\right)a=\frac{b^{2}}{24}$$

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

$$\frac{\left(-\frac{b}{2}\right)a}{-\frac{b}{2}}=\frac{b^{2}}{24\left(-\frac{b}{2}\right)}$$

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

$$a=\frac{b^{2}}{24\left(-\frac{b}{2}\right)}$$

Divide \frac{b^{2}}{24} by -\frac{1}{2}b.

$$a=-\frac{b}{12}$$

Solve for b

$$b=-12a$$

$$b=0$$