Solve {l}{2x+3/3y-2=1}{x(2y+2)-2y(x+3)=2x+1}

$$\left. \begin{array} { l } { \frac { 2 x + 3 } { 3 y - 2 } = 1 } \\ { x ( 2 y + 2 ) - 2 y ( x + 3 ) = 2 x + 1 } \end{array} \right.$$

Solve for x, y

$$x = -\frac{11}{4} = -2\frac{3}{4} = -2.75$$

$$y=-\frac{1}{6}\approx -0.166666667$$

Solution Steps

Consider the first equation. Variable y cannot be equal to \frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by 3y-2.

$$2x+3=3y-2$$

Subtract 3y from both sides.

$$2x+3-3y=-2$$

Subtract 3 from both sides.

$$2x-3y=-2-3$$

Subtract 3 from -2 to get -5.

$$2x-3y=-5$$

Consider the second equation. Use the distributive property to multiply x by 2y+2.

$$2xy+2x-2y\left(x+3\right)=2x+1$$

Subtract 2x from both sides.

$$2xy+2x-2y\left(x+3\right)-2x=1$$

Use the distributive property to multiply -2y by x+3.

$$2xy+2x-2yx-6y-2x=1$$

Combine 2xy and -2yx to get 0.

$$2x-6y-2x=1$$

Combine 2x and -2x to get 0.

$$-6y=1$$

Divide both sides by -6.

$$y=-\frac{1}{6}$$

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

$$2x-3\left(-\frac{1}{6}\right)=-5$$

Multiply -3 and -\frac{1}{6} to get \frac{1}{2}.

$$2x+\frac{1}{2}=-5$$

Subtract \frac{1}{2} from both sides.

$$2x=-5-\frac{1}{2}$$

Subtract \frac{1}{2} from -5 to get -\frac{11}{2}.

$$2x=-\frac{11}{2}$$

Divide both sides by 2.

$$x=\frac{-\frac{11}{2}}{2}$$

Express \frac{-\frac{11}{2}}{2} as a single fraction.

$$x=\frac{-11}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$x=\frac{-11}{4}$$

Fraction \frac{-11}{4} can be rewritten as -\frac{11}{4} by extracting the negative sign.

$$x=-\frac{11}{4}$$

The system is now solved.

$$x=-\frac{11}{4} y=-\frac{1}{6}$$