Solve 3/4(y+7)+1/2(3y-5)=9/4(2y-1)

$$\frac { 3 } { 4 } ( y + 7 ) + \frac { 1 } { 2 } ( 3 y - 5 ) = \frac { 9 } { 4 } ( 2 y - 1 )$$

Solve for y

$$y = \frac{20}{9} = 2\frac{2}{9} \approx 2.222222222$$

Steps for Solving Linear Equation

Use the distributive property to multiply \frac{3}{4} by y+7.

$$\frac{3}{4}y+\frac{3}{4}\times 7+\frac{1}{2}\left(3y-5\right)=\frac{9}{4}\left(2y-1\right)$$

Express \frac{3}{4}\times 7 as a single fraction.

$$\frac{3}{4}y+\frac{3\times 7}{4}+\frac{1}{2}\left(3y-5\right)=\frac{9}{4}\left(2y-1\right)$$

Multiply 3 and 7 to get 21.

$$\frac{3}{4}y+\frac{21}{4}+\frac{1}{2}\left(3y-5\right)=\frac{9}{4}\left(2y-1\right)$$

Use the distributive property to multiply \frac{1}{2} by 3y-5.

$$\frac{3}{4}y+\frac{21}{4}+\frac{1}{2}\times 3y+\frac{1}{2}\left(-5\right)=\frac{9}{4}\left(2y-1\right)$$

Multiply \frac{1}{2} and 3 to get \frac{3}{2}.

$$\frac{3}{4}y+\frac{21}{4}+\frac{3}{2}y+\frac{1}{2}\left(-5\right)=\frac{9}{4}\left(2y-1\right)$$

Multiply \frac{1}{2} and -5 to get \frac{-5}{2}.

$$\frac{3}{4}y+\frac{21}{4}+\frac{3}{2}y+\frac{-5}{2}=\frac{9}{4}\left(2y-1\right)$$

Fraction \frac{-5}{2} can be rewritten as -\frac{5}{2} by extracting the negative sign.

$$\frac{3}{4}y+\frac{21}{4}+\frac{3}{2}y-\frac{5}{2}=\frac{9}{4}\left(2y-1\right)$$

Combine \frac{3}{4}y and \frac{3}{2}y to get \frac{9}{4}y.

$$\frac{9}{4}y+\frac{21}{4}-\frac{5}{2}=\frac{9}{4}\left(2y-1\right)$$

Least common multiple of 4 and 2 is 4. Convert \frac{21}{4} and \frac{5}{2} to fractions with denominator 4.

$$\frac{9}{4}y+\frac{21}{4}-\frac{10}{4}=\frac{9}{4}\left(2y-1\right)$$

Since \frac{21}{4} and \frac{10}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{9}{4}y+\frac{21-10}{4}=\frac{9}{4}\left(2y-1\right)$$

Subtract 10 from 21 to get 11.

$$\frac{9}{4}y+\frac{11}{4}=\frac{9}{4}\left(2y-1\right)$$

Use the distributive property to multiply \frac{9}{4} by 2y-1.

$$\frac{9}{4}y+\frac{11}{4}=\frac{9}{4}\times 2y+\frac{9}{4}\left(-1\right)$$

Express \frac{9}{4}\times 2 as a single fraction.

$$\frac{9}{4}y+\frac{11}{4}=\frac{9\times 2}{4}y+\frac{9}{4}\left(-1\right)$$

Multiply 9 and 2 to get 18.

$$\frac{9}{4}y+\frac{11}{4}=\frac{18}{4}y+\frac{9}{4}\left(-1\right)$$

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

$$\frac{9}{4}y+\frac{11}{4}=\frac{9}{2}y+\frac{9}{4}\left(-1\right)$$

Multiply \frac{9}{4} and -1 to get -\frac{9}{4}.

$$\frac{9}{4}y+\frac{11}{4}=\frac{9}{2}y-\frac{9}{4}$$

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

$$\frac{9}{4}y+\frac{11}{4}-\frac{9}{2}y=-\frac{9}{4}$$

Combine \frac{9}{4}y and -\frac{9}{2}y to get -\frac{9}{4}y.

$$-\frac{9}{4}y+\frac{11}{4}=-\frac{9}{4}$$

Subtract \frac{11}{4} from both sides.

$$-\frac{9}{4}y=-\frac{9}{4}-\frac{11}{4}$$

Since -\frac{9}{4} and \frac{11}{4} have the same denominator, subtract them by subtracting their numerators.

$$-\frac{9}{4}y=\frac{-9-11}{4}$$

Subtract 11 from -9 to get -20.

$$-\frac{9}{4}y=\frac{-20}{4}$$

Divide -20 by 4 to get -5.

$$-\frac{9}{4}y=-5$$

Multiply both sides by -\frac{4}{9}, the reciprocal of -\frac{9}{4}.

$$y=-5\left(-\frac{4}{9}\right)$$

Express -5\left(-\frac{4}{9}\right) as a single fraction.

$$y=\frac{-5\left(-4\right)}{9}$$

Multiply -5 and -4 to get 20.

$$y=\frac{20}{9}$$