Solve r/2(3/4-4r)=(r/4-1/2)(frac{1}{2}-8r)

$$\frac { r } { 2 } ( \frac { 3 } { 4 } - 4 r ) = ( \frac { r } { 4 } - \frac { 1 } { 2 } ) ( \frac { 1 } { 2 } - 8 r )$$

Solve for r

$$r=\frac{1}{15}\approx 0.066666667$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$2r\left(\frac{3}{4}-4r\right)=4\left(\frac{r}{4}-\frac{1}{2}\right)\left(\frac{1}{2}-8r\right)$$

Use the distributive property to multiply 2r by \frac{3}{4}-4r.

$$2r\times \frac{3}{4}-8r^{2}=4\left(\frac{r}{4}-\frac{1}{2}\right)\left(\frac{1}{2}-8r\right)$$

Express 2\times \frac{3}{4} as a single fraction.

$$\frac{2\times 3}{4}r-8r^{2}=4\left(\frac{r}{4}-\frac{1}{2}\right)\left(\frac{1}{2}-8r\right)$$

Multiply 2 and 3 to get 6.

$$\frac{6}{4}r-8r^{2}=4\left(\frac{r}{4}-\frac{1}{2}\right)\left(\frac{1}{2}-8r\right)$$

Reduce the fraction \frac{6}{4} to lowest terms by extracting and canceling out 2.

$$\frac{3}{2}r-8r^{2}=4\left(\frac{r}{4}-\frac{1}{2}\right)\left(\frac{1}{2}-8r\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 2 is 4. Multiply \frac{1}{2} times \frac{2}{2}.

$$\frac{3}{2}r-8r^{2}=4\left(\frac{r}{4}-\frac{2}{4}\right)\left(\frac{1}{2}-8r\right)$$

Since \frac{r}{4} and \frac{2}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3}{2}r-8r^{2}=4\times \frac{r-2}{4}\left(\frac{1}{2}-8r\right)$$

Express 4\times \frac{r-2}{4} as a single fraction.

$$\frac{3}{2}r-8r^{2}=\frac{4\left(r-2\right)}{4}\left(\frac{1}{2}-8r\right)$$

Cancel out 4 and 4.

$$\frac{3}{2}r-8r^{2}=\left(r-2\right)\left(\frac{1}{2}-8r\right)$$

Apply the distributive property by multiplying each term of r-2 by each term of \frac{1}{2}-8r.

$$\frac{3}{2}r-8r^{2}=r\times \frac{1}{2}-8r^{2}-2\times \frac{1}{2}+16r$$

Multiply -2 times \frac{1}{2}.

$$\frac{3}{2}r-8r^{2}=r\times \frac{1}{2}-8r^{2}-1+16r$$

Combine r\times \frac{1}{2} and 16r to get \frac{33}{2}r.

$$\frac{3}{2}r-8r^{2}=\frac{33}{2}r-8r^{2}-1$$

Subtract \frac{33}{2}r from both sides.

$$\frac{3}{2}r-8r^{2}-\frac{33}{2}r=-8r^{2}-1$$

Combine \frac{3}{2}r and -\frac{33}{2}r to get -15r.

$$-15r-8r^{2}=-8r^{2}-1$$

Add 8r^{2} to both sides.

$$-15r-8r^{2}+8r^{2}=-1$$

Combine -8r^{2} and 8r^{2} to get 0.

$$-15r=-1$$

Divide both sides by -15.

$$r=\frac{-1}{-15}$$

Fraction \frac{-1}{-15} can be simplified to \frac{1}{15} by removing the negative sign from both the numerator and the denominator.

$$r=\frac{1}{15}$$