Solve {r}{3/3+4y=10/3}{2(-2y+x)/3-3x/2=-frac{13}{6}}

$$\left. \begin{array} { r } { \frac { 3 } { 3 } + 4 y = \frac { 10 } { 3 } } \\ { \frac { 2 ( - 2 y + x ) } { 3 } - \frac { 3 x } { 2 } = - \frac { 13 } { 6 } } \end{array} \right.$$

Solve for y, x

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

$$y=\frac{7}{12}\approx 0.583333333$$

Solution Steps

Consider the first equation. Divide 3 by 3 to get 1.

$$1+4y=\frac{10}{3}$$

Subtract 1 from both sides.

$$4y=\frac{10}{3}-1$$

Subtract 1 from \frac{10}{3} to get \frac{7}{3}.

$$4y=\frac{7}{3}$$

Divide both sides by 4.

$$y=\frac{\frac{7}{3}}{4}$$

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

$$y=\frac{7}{3\times 4}$$

Multiply 3 and 4 to get 12.

$$y=\frac{7}{12}$$

Consider the second equation. Insert the known values of variables into the equation.

$$\frac{2\left(-2\times \frac{7}{12}+x\right)}{3}-\frac{3x}{2}=-\frac{13}{6}$$

Multiply both sides of the equation by 6, the least common multiple of 3,2,6.

$$2\times 2\left(-2\times \frac{7}{12}+x\right)-3\times 3x=-13$$

Multiply 2 and 2 to get 4.

$$4\left(-2\times \frac{7}{12}+x\right)-3\times 3x=-13$$

Multiply -2 and \frac{7}{12} to get -\frac{7}{6}.

$$4\left(-\frac{7}{6}+x\right)-3\times 3x=-13$$

Use the distributive property to multiply 4 by -\frac{7}{6}+x.

$$-\frac{14}{3}+4x-3\times 3x=-13$$

Multiply -3 and 3 to get -9.

$$-\frac{14}{3}+4x-9x=-13$$

Combine 4x and -9x to get -5x.

$$-\frac{14}{3}-5x=-13$$

Add \frac{14}{3} to both sides.

$$-5x=-13+\frac{14}{3}$$

Add -13 and \frac{14}{3} to get -\frac{25}{3}.

$$-5x=-\frac{25}{3}$$

Divide both sides by -5.

$$x=\frac{-\frac{25}{3}}{-5}$$

Express \frac{-\frac{25}{3}}{-5} as a single fraction.

$$x=\frac{-25}{3\left(-5\right)}$$

Multiply 3 and -5 to get -15.

$$x=\frac{-25}{-15}$$

Reduce the fraction \frac{-25}{-15} to lowest terms by extracting and canceling out -5.

$$x=\frac{5}{3}$$

The system is now solved.

$$y=\frac{7}{12} x=\frac{5}{3}$$