Solve {l}{3/8(2w+3)+5/4w=3/4(4w+1)}{3/4(y+7)+frac{1}{2}(3y-5)=frac{9}{4}(2y-1)}

$$\left. \begin{array} { l } { \frac { 3 } { 8 } ( 2 w + 3 ) + \frac { 5 } { 4 } w = \frac { 3 } { 4 } ( 4 w + 1 ) } \\ { \frac { 3 } { 4 } ( y + 7 ) + \frac { 1 } { 2 } ( 3 y - 5 ) = \frac { 9 } { 4 } ( 2 y - 1 ) } \end{array} \right.$$

Solve for w, y

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

$$w=\frac{3}{8}=0.375$$

Solution Steps

Consider the first equation. Use the distributive property to multiply \frac{3}{8} by 2w+3.

$$\frac{3}{4}w+\frac{9}{8}+\frac{5}{4}w=\frac{3}{4}\left(4w+1\right)$$

Combine \frac{3}{4}w and \frac{5}{4}w to get 2w.

$$2w+\frac{9}{8}=\frac{3}{4}\left(4w+1\right)$$

Use the distributive property to multiply \frac{3}{4} by 4w+1.

$$2w+\frac{9}{8}=3w+\frac{3}{4}$$

Subtract 3w from both sides.

$$2w+\frac{9}{8}-3w=\frac{3}{4}$$

Combine 2w and -3w to get -w.

$$-w+\frac{9}{8}=\frac{3}{4}$$

Subtract \frac{9}{8} from both sides.

$$-w=\frac{3}{4}-\frac{9}{8}$$

Subtract \frac{9}{8} from \frac{3}{4} to get -\frac{3}{8}.

$$-w=-\frac{3}{8}$$

Divide both sides by -1.

$$w=\frac{-\frac{3}{8}}{-1}$$

Express \frac{-\frac{3}{8}}{-1} as a single fraction.

$$w=\frac{-3}{8\left(-1\right)}$$

Multiply 8 and -1 to get -8.

$$w=\frac{-3}{-8}$$

Fraction \frac{-3}{-8} can be simplified to \frac{3}{8} by removing the negative sign from both the numerator and the denominator.

$$w=\frac{3}{8}$$

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

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

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

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

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

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

Subtract \frac{5}{2} from \frac{21}{4} to get \frac{11}{4}.

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

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

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

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.

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

Subtract \frac{11}{4} from -\frac{9}{4} to get -5.

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

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

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

Multiply -5 and -\frac{4}{9} to get \frac{20}{9}.

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

The system is now solved.

$$w=\frac{3}{8} y=\frac{20}{9}$$