Solve {l}{2/3x+[(x+2y)(x-1)-x^2+x]=1/2(1+y/2)+y(2x-2)}{2y-13/3x=-3}

$$\left\{ \begin{array} { l } { \frac { 2 } { 3 } x + [ ( x + 2 y ) ( x - 1 ) - x ^ { 2 } + x ] = \frac { 1 } { 2 } ( 1 + \frac { y } { 2 } ) + y ( 2 x - 2 ) } \\ { 2 y - \frac { 13 } { 3 } x = - 3 } \end{array} \right.$$

Solve for x, y

$$x=1$$

$$y=\frac{2}{3}\approx 0.666666667$$

Steps Using Substitution

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

$$4x+6\left(x+2y\right)\left(x-1\right)-6x^{2}+6x=3\left(1+\frac{y}{2}\right)+6y\left(2x-2\right)$$

Use the distributive property to multiply 6 by x+2y.

$$4x+\left(6x+12y\right)\left(x-1\right)-6x^{2}+6x=3\left(1+\frac{y}{2}\right)+6y\left(2x-2\right)$$

Use the distributive property to multiply 6x+12y by x-1.

$$4x+6x^{2}-6x+12yx-12y-6x^{2}+6x=3\left(1+\frac{y}{2}\right)+6y\left(2x-2\right)$$

Combine 4x and -6x to get -2x.

$$-2x+6x^{2}+12yx-12y-6x^{2}+6x=3\left(1+\frac{y}{2}\right)+6y\left(2x-2\right)$$

Combine 6x^{2} and -6x^{2} to get 0.

$$-2x+12yx-12y+6x=3\left(1+\frac{y}{2}\right)+6y\left(2x-2\right)$$

Combine -2x and 6x to get 4x.

$$4x+12yx-12y=3\left(1+\frac{y}{2}\right)+6y\left(2x-2\right)$$

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

$$4x+12yx-12y=3+3\times \frac{y}{2}+6y\left(2x-2\right)$$

Express 3\times \frac{y}{2} as a single fraction.

$$4x+12yx-12y=3+\frac{3y}{2}+6y\left(2x-2\right)$$

Use the distributive property to multiply 6y by 2x-2.

$$4x+12yx-12y=3+\frac{3y}{2}+12yx-12y$$

Subtract \frac{3y}{2} from both sides.

$$4x+12yx-12y-\frac{3y}{2}=3+12yx-12y$$

Subtract 12yx from both sides.

$$4x+12yx-12y-\frac{3y}{2}-12yx=3-12y$$

Combine 12yx and -12yx to get 0.

$$4x-12y-\frac{3y}{2}=3-12y$$

Add 12y to both sides.

$$4x-12y-\frac{3y}{2}+12y=3$$

Combine -12y and 12y to get 0.

$$4x-\frac{3y}{2}=3$$

Multiply both sides of the equation by 2.

$$8x-3y=6$$

To solve a pair of equations using substitution, first solve one of the equations for one of the variables. Then substitute the result for that variable in the other equation.

$$8x-3y=6,-\frac{13}{3}x+2y=-3$$

Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.

$$8x-3y=6$$

Add 3y to both sides of the equation.

$$8x=3y+6$$

Divide both sides by 8.

$$x=\frac{1}{8}\left(3y+6\right)$$

Multiply \frac{1}{8} times 6+3y.

$$x=\frac{3}{8}y+\frac{3}{4}$$

Substitute \frac{3}{4}+\frac{3y}{8} for x in the other equation, -\frac{13}{3}x+2y=-3.

$$-\frac{13}{3}\left(\frac{3}{8}y+\frac{3}{4}\right)+2y=-3$$

Multiply -\frac{13}{3} times \frac{3}{4}+\frac{3y}{8}.

$$-\frac{13}{8}y-\frac{13}{4}+2y=-3$$

Add -\frac{13y}{8} to 2y.

$$\frac{3}{8}y-\frac{13}{4}=-3$$

Add \frac{13}{4} to both sides of the equation.

$$\frac{3}{8}y=\frac{1}{4}$$

Divide both sides of the equation by \frac{3}{8}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{2}{3}$$

Substitute \frac{2}{3} for y in x=\frac{3}{8}y+\frac{3}{4}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{3}{8}\times \frac{2}{3}+\frac{3}{4}$$

Multiply \frac{3}{8} times \frac{2}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{1+3}{4}$$

Add \frac{3}{4} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=1$$

The system is now solved.

$$x=1,y=\frac{2}{3}$$

Steps Using Matrices

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

$$4x+6\left(x+2y\right)\left(x-1\right)-6x^{2}+6x=3\left(1+\frac{y}{2}\right)+6y\left(2x-2\right)$$

Use the distributive property to multiply 6 by x+2y.

$$4x+\left(6x+12y\right)\left(x-1\right)-6x^{2}+6x=3\left(1+\frac{y}{2}\right)+6y\left(2x-2\right)$$

Use the distributive property to multiply 6x+12y by x-1.

$$4x+6x^{2}-6x+12yx-12y-6x^{2}+6x=3\left(1+\frac{y}{2}\right)+6y\left(2x-2\right)$$

Combine 4x and -6x to get -2x.

$$-2x+6x^{2}+12yx-12y-6x^{2}+6x=3\left(1+\frac{y}{2}\right)+6y\left(2x-2\right)$$

Combine 6x^{2} and -6x^{2} to get 0.

$$-2x+12yx-12y+6x=3\left(1+\frac{y}{2}\right)+6y\left(2x-2\right)$$

Combine -2x and 6x to get 4x.

$$4x+12yx-12y=3\left(1+\frac{y}{2}\right)+6y\left(2x-2\right)$$

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

$$4x+12yx-12y=3+3\times \frac{y}{2}+6y\left(2x-2\right)$$

Express 3\times \frac{y}{2} as a single fraction.

$$4x+12yx-12y=3+\frac{3y}{2}+6y\left(2x-2\right)$$

Use the distributive property to multiply 6y by 2x-2.

$$4x+12yx-12y=3+\frac{3y}{2}+12yx-12y$$

Subtract \frac{3y}{2} from both sides.

$$4x+12yx-12y-\frac{3y}{2}=3+12yx-12y$$

Subtract 12yx from both sides.

$$4x+12yx-12y-\frac{3y}{2}-12yx=3-12y$$

Combine 12yx and -12yx to get 0.

$$4x-12y-\frac{3y}{2}=3-12y$$

Add 12y to both sides.

$$4x-12y-\frac{3y}{2}+12y=3$$

Combine -12y and 12y to get 0.

$$4x-\frac{3y}{2}=3$$

Multiply both sides of the equation by 2.

$$8x-3y=6$$

Put the equations in standard form and then use matrices to solve the system of equations.

$$8x-3y=6,-\frac{13}{3}x+2y=-3$$

Write the equations in matrix form.

$$\left(\begin{matrix}8&-3\\-\frac{13}{3}&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\-3\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}8&-3\\-\frac{13}{3}&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&-3\\-\frac{13}{3}&2\end{matrix}\right))\left(\begin{matrix}8&-3\\-\frac{13}{3}&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-3\\-\frac{13}{3}&2\end{matrix}\right))\left(\begin{matrix}6\\-3\end{matrix}\right)$$

The product of a matrix and its inverse is the identity matrix.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-3\\-\frac{13}{3}&2\end{matrix}\right))\left(\begin{matrix}6\\-3\end{matrix}\right)$$

Multiply the matrices on the left hand side of the equal sign.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-3\\-\frac{13}{3}&2\end{matrix}\right))\left(\begin{matrix}6\\-3\end{matrix}\right)$$

For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the inverse matrix is \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), so the matrix equation can be rewritten as a matrix multiplication problem.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{8\times 2-\left(-3\left(-\frac{13}{3}\right)\right)}&-\frac{-3}{8\times 2-\left(-3\left(-\frac{13}{3}\right)\right)}\\-\frac{-\frac{13}{3}}{8\times 2-\left(-3\left(-\frac{13}{3}\right)\right)}&\frac{8}{8\times 2-\left(-3\left(-\frac{13}{3}\right)\right)}\end{matrix}\right)\left(\begin{matrix}6\\-3\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}&1\\\frac{13}{9}&\frac{8}{3}\end{matrix}\right)\left(\begin{matrix}6\\-3\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}\times 6-3\\\frac{13}{9}\times 6+\frac{8}{3}\left(-3\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\\frac{2}{3}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=1,y=\frac{2}{3}$$

Steps Using Elimination

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

$$4x+6\left(x+2y\right)\left(x-1\right)-6x^{2}+6x=3\left(1+\frac{y}{2}\right)+6y\left(2x-2\right)$$

Use the distributive property to multiply 6 by x+2y.

$$4x+\left(6x+12y\right)\left(x-1\right)-6x^{2}+6x=3\left(1+\frac{y}{2}\right)+6y\left(2x-2\right)$$

Use the distributive property to multiply 6x+12y by x-1.

$$4x+6x^{2}-6x+12yx-12y-6x^{2}+6x=3\left(1+\frac{y}{2}\right)+6y\left(2x-2\right)$$

Combine 4x and -6x to get -2x.

$$-2x+6x^{2}+12yx-12y-6x^{2}+6x=3\left(1+\frac{y}{2}\right)+6y\left(2x-2\right)$$

Combine 6x^{2} and -6x^{2} to get 0.

$$-2x+12yx-12y+6x=3\left(1+\frac{y}{2}\right)+6y\left(2x-2\right)$$

Combine -2x and 6x to get 4x.

$$4x+12yx-12y=3\left(1+\frac{y}{2}\right)+6y\left(2x-2\right)$$

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

$$4x+12yx-12y=3+3\times \frac{y}{2}+6y\left(2x-2\right)$$

Express 3\times \frac{y}{2} as a single fraction.

$$4x+12yx-12y=3+\frac{3y}{2}+6y\left(2x-2\right)$$

Use the distributive property to multiply 6y by 2x-2.

$$4x+12yx-12y=3+\frac{3y}{2}+12yx-12y$$

Subtract \frac{3y}{2} from both sides.

$$4x+12yx-12y-\frac{3y}{2}=3+12yx-12y$$

Subtract 12yx from both sides.

$$4x+12yx-12y-\frac{3y}{2}-12yx=3-12y$$

Combine 12yx and -12yx to get 0.

$$4x-12y-\frac{3y}{2}=3-12y$$

Add 12y to both sides.

$$4x-12y-\frac{3y}{2}+12y=3$$

Combine -12y and 12y to get 0.

$$4x-\frac{3y}{2}=3$$

Multiply both sides of the equation by 2.

$$8x-3y=6$$

In order to solve by elimination, coefficients of one of the variables must be the same in both equations so that the variable will cancel out when one equation is subtracted from the other.

$$8x-3y=6,-\frac{13}{3}x+2y=-3$$

To make 8x and -\frac{13x}{3} equal, multiply all terms on each side of the first equation by -\frac{13}{3} and all terms on each side of the second by 8.

$$-\frac{13}{3}\times 8x-\frac{13}{3}\left(-3\right)y=-\frac{13}{3}\times 6,8\left(-\frac{13}{3}\right)x+8\times 2y=8\left(-3\right)$$

Simplify.

$$-\frac{104}{3}x+13y=-26,-\frac{104}{3}x+16y=-24$$

Subtract -\frac{104}{3}x+16y=-24 from -\frac{104}{3}x+13y=-26 by subtracting like terms on each side of the equal sign.

$$-\frac{104}{3}x+\frac{104}{3}x+13y-16y=-26+24$$

Add -\frac{104x}{3} to \frac{104x}{3}. Terms -\frac{104x}{3} and \frac{104x}{3} cancel out, leaving an equation with only one variable that can be solved.

$$13y-16y=-26+24$$

Add 13y to -16y.

$$-3y=-26+24$$

Add -26 to 24.

$$-3y=-2$$

Divide both sides by -3.

$$y=\frac{2}{3}$$

Substitute \frac{2}{3} for y in -\frac{13}{3}x+2y=-3. Because the resulting equation contains only one variable, you can solve for x directly.

$$-\frac{13}{3}x+2\times \frac{2}{3}=-3$$

Multiply 2 times \frac{2}{3}.

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

Subtract \frac{4}{3} from both sides of the equation.

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

Divide both sides of the equation by -\frac{13}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=1$$

The system is now solved.

$$x=1,y=\frac{2}{3}$$