Solve 1/2[1+1/4(3z-1)]=2z/3-1/2

$$\frac { 1 } { 2 } [ 1 + \frac { 1 } { 4 } ( 3 z - 1 ) ] = \frac { 2 z } { 3 } - \frac { 1 } { 2 }$$

Solve for z

$$z=3$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 12, the least common multiple of 2,4,3.

$$6\left(1+\frac{1}{4}\left(3z-1\right)\right)=4\times 2z-6$$

Use the distributive property to multiply \frac{1}{4} by 3z-1.

$$6\left(1+\frac{1}{4}\times 3z+\frac{1}{4}\left(-1\right)\right)=4\times 2z-6$$

Multiply \frac{1}{4} and 3 to get \frac{3}{4}.

$$6\left(1+\frac{3}{4}z+\frac{1}{4}\left(-1\right)\right)=4\times 2z-6$$

Multiply \frac{1}{4} and -1 to get -\frac{1}{4}.

$$6\left(1+\frac{3}{4}z-\frac{1}{4}\right)=4\times 2z-6$$

Convert 1 to fraction \frac{4}{4}.

$$6\left(\frac{4}{4}+\frac{3}{4}z-\frac{1}{4}\right)=4\times 2z-6$$

Since \frac{4}{4} and \frac{1}{4} have the same denominator, subtract them by subtracting their numerators.

$$6\left(\frac{4-1}{4}+\frac{3}{4}z\right)=4\times 2z-6$$

Subtract 1 from 4 to get 3.

$$6\left(\frac{3}{4}+\frac{3}{4}z\right)=4\times 2z-6$$

Use the distributive property to multiply 6 by \frac{3}{4}+\frac{3}{4}z.

$$6\times \frac{3}{4}+6\times \frac{3}{4}z=4\times 2z-6$$

Express 6\times \frac{3}{4} as a single fraction.

$$\frac{6\times 3}{4}+6\times \frac{3}{4}z=4\times 2z-6$$

Multiply 6 and 3 to get 18.

$$\frac{18}{4}+6\times \frac{3}{4}z=4\times 2z-6$$

Reduce the fraction \frac{18}{4} to lowest terms by extracting and canceling out 2.

$$\frac{9}{2}+6\times \frac{3}{4}z=4\times 2z-6$$

Express 6\times \frac{3}{4} as a single fraction.

$$\frac{9}{2}+\frac{6\times 3}{4}z=4\times 2z-6$$

Multiply 6 and 3 to get 18.

$$\frac{9}{2}+\frac{18}{4}z=4\times 2z-6$$

Reduce the fraction \frac{18}{4} to lowest terms by extracting and canceling out 2.

$$\frac{9}{2}+\frac{9}{2}z=4\times 2z-6$$

Multiply 4 and 2 to get 8.

$$\frac{9}{2}+\frac{9}{2}z=8z-6$$

Subtract 8z from both sides.

$$\frac{9}{2}+\frac{9}{2}z-8z=-6$$

Combine \frac{9}{2}z and -8z to get -\frac{7}{2}z.

$$\frac{9}{2}-\frac{7}{2}z=-6$$

Subtract \frac{9}{2} from both sides.

$$-\frac{7}{2}z=-6-\frac{9}{2}$$

Convert -6 to fraction -\frac{12}{2}.

$$-\frac{7}{2}z=-\frac{12}{2}-\frac{9}{2}$$

Since -\frac{12}{2} and \frac{9}{2} have the same denominator, subtract them by subtracting their numerators.

$$-\frac{7}{2}z=\frac{-12-9}{2}$$

Subtract 9 from -12 to get -21.

$$-\frac{7}{2}z=-\frac{21}{2}$$

Multiply both sides by -\frac{2}{7}, the reciprocal of -\frac{7}{2}.

$$z=-\frac{21}{2}\left(-\frac{2}{7}\right)$$

Multiply -\frac{21}{2} times -\frac{2}{7} by multiplying numerator times numerator and denominator times denominator.

$$z=\frac{-21\left(-2\right)}{2\times 7}$$

Do the multiplications in the fraction \frac{-21\left(-2\right)}{2\times 7}.

$$z=\frac{42}{14}$$

Divide 42 by 14 to get 3.

$$z=3$$