Solve {l}{y(y-2x)+(x-y)(x+y)+3x-2y(1-x)=-11/2+x^2}{4[y+1/2-(x+2)]+x-7/3+y+2/2=x}

$$\left\{ \begin{array} { l } { y ( y - 2 x ) + ( x - y ) ( x + y ) + 3 x - 2 y ( 1 - x ) = - \frac { 11 } { 2 } + x ^ { 2 } } \\ { 4 [ \frac { y + 1 } { 2 } - ( x + 2 ) ] + \frac { x - 7 } { 3 } + \frac { y + 2 } { 2 } = x } \end{array} \right.$$

Solve for y, x

$$x=-\frac{1}{2}=-0.5$$

$$y=2$$

Steps Using Substitution

Consider the first equation. Use the distributive property to multiply y by y-2x.

$$y^{2}-2yx+\left(x-y\right)\left(x+y\right)+3x-2y\left(1-x\right)=-\frac{11}{2}+x^{2}$$

Consider \left(x-y\right)\left(x+y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$y^{2}-2yx+x^{2}-y^{2}+3x-2y\left(1-x\right)=-\frac{11}{2}+x^{2}$$

Combine y^{2} and -y^{2} to get 0.

$$-2yx+x^{2}+3x-2y\left(1-x\right)=-\frac{11}{2}+x^{2}$$

Subtract x^{2} from both sides.

$$-2yx+x^{2}+3x-2y\left(1-x\right)-x^{2}=-\frac{11}{2}$$

Use the distributive property to multiply -2y by 1-x.

$$-2yx+x^{2}+3x-2y+2yx-x^{2}=-\frac{11}{2}$$

Combine -2yx and 2yx to get 0.

$$x^{2}+3x-2y-x^{2}=-\frac{11}{2}$$

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

$$3x-2y=-\frac{11}{2}$$

Consider the second equation. To find the opposite of x+2, find the opposite of each term.

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

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

$$4\times \frac{y+1}{2}-4x-8+\frac{x-7}{3}+\frac{y+2}{2}=x$$

Cancel out 2, the greatest common factor in 4 and 2.

$$2\left(y+1\right)-4x-8+\frac{x-7}{3}+\frac{y+2}{2}=x$$

Use the distributive property to multiply 2 by y+1.

$$2y+2-4x-8+\frac{x-7}{3}+\frac{y+2}{2}=x$$

Subtract 8 from 2 to get -6.

$$2y-6-4x+\frac{x-7}{3}+\frac{y+2}{2}=x$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{x-7}{3} times \frac{2}{2}. Multiply \frac{y+2}{2} times \frac{3}{3}.

$$2y-6-4x+\frac{2\left(x-7\right)}{6}+\frac{3\left(y+2\right)}{6}=x$$

Since \frac{2\left(x-7\right)}{6} and \frac{3\left(y+2\right)}{6} have the same denominator, add them by adding their numerators.

$$2y-6-4x+\frac{2\left(x-7\right)+3\left(y+2\right)}{6}=x$$

Do the multiplications in 2\left(x-7\right)+3\left(y+2\right).

$$2y-6-4x+\frac{2x-14+3y+6}{6}=x$$

Combine like terms in 2x-14+3y+6.

$$2y-6-4x+\frac{2x-8+3y}{6}=x$$

Divide each term of 2x-8+3y by 6 to get \frac{1}{3}x-\frac{4}{3}+\frac{1}{2}y.

$$2y-6-4x+\frac{1}{3}x-\frac{4}{3}+\frac{1}{2}y=x$$

Combine -4x and \frac{1}{3}x to get -\frac{11}{3}x.

$$2y-6-\frac{11}{3}x-\frac{4}{3}+\frac{1}{2}y=x$$

Subtract \frac{4}{3} from -6 to get -\frac{22}{3}.

$$2y-\frac{22}{3}-\frac{11}{3}x+\frac{1}{2}y=x$$

Combine 2y and \frac{1}{2}y to get \frac{5}{2}y.

$$\frac{5}{2}y-\frac{22}{3}-\frac{11}{3}x=x$$

Subtract x from both sides.

$$\frac{5}{2}y-\frac{22}{3}-\frac{11}{3}x-x=0$$

Combine -\frac{11}{3}x and -x to get -\frac{14}{3}x.

$$\frac{5}{2}y-\frac{22}{3}-\frac{14}{3}x=0$$

Add \frac{22}{3} to both sides. Anything plus zero gives itself.

$$\frac{5}{2}y-\frac{14}{3}x=\frac{22}{3}$$

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.

$$3x-2y=-\frac{11}{2},-\frac{14}{3}x+\frac{5}{2}y=\frac{22}{3}$$

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

$$3x-2y=-\frac{11}{2}$$

Add 2y to both sides of the equation.

$$3x=2y-\frac{11}{2}$$

Divide both sides by 3.

$$x=\frac{1}{3}\left(2y-\frac{11}{2}\right)$$

Multiply \frac{1}{3} times 2y-\frac{11}{2}.

$$x=\frac{2}{3}y-\frac{11}{6}$$

Substitute \frac{2y}{3}-\frac{11}{6} for x in the other equation, -\frac{14}{3}x+\frac{5}{2}y=\frac{22}{3}.

$$-\frac{14}{3}\left(\frac{2}{3}y-\frac{11}{6}\right)+\frac{5}{2}y=\frac{22}{3}$$

Multiply -\frac{14}{3} times \frac{2y}{3}-\frac{11}{6}.

$$-\frac{28}{9}y+\frac{77}{9}+\frac{5}{2}y=\frac{22}{3}$$

Add -\frac{28y}{9} to \frac{5y}{2}.

$$-\frac{11}{18}y+\frac{77}{9}=\frac{22}{3}$$

Subtract \frac{77}{9} from both sides of the equation.

$$-\frac{11}{18}y=-\frac{11}{9}$$

Divide both sides of the equation by -\frac{11}{18}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=2$$

Substitute 2 for y in x=\frac{2}{3}y-\frac{11}{6}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{2}{3}\times 2-\frac{11}{6}$$

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

$$x=\frac{4}{3}-\frac{11}{6}$$

Add -\frac{11}{6} to \frac{4}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

The system is now solved.

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

Steps Using Matrices

Consider the first equation. Use the distributive property to multiply y by y-2x.

$$y^{2}-2yx+\left(x-y\right)\left(x+y\right)+3x-2y\left(1-x\right)=-\frac{11}{2}+x^{2}$$

Consider \left(x-y\right)\left(x+y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$y^{2}-2yx+x^{2}-y^{2}+3x-2y\left(1-x\right)=-\frac{11}{2}+x^{2}$$

Combine y^{2} and -y^{2} to get 0.

$$-2yx+x^{2}+3x-2y\left(1-x\right)=-\frac{11}{2}+x^{2}$$

Subtract x^{2} from both sides.

$$-2yx+x^{2}+3x-2y\left(1-x\right)-x^{2}=-\frac{11}{2}$$

Use the distributive property to multiply -2y by 1-x.

$$-2yx+x^{2}+3x-2y+2yx-x^{2}=-\frac{11}{2}$$

Combine -2yx and 2yx to get 0.

$$x^{2}+3x-2y-x^{2}=-\frac{11}{2}$$

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

$$3x-2y=-\frac{11}{2}$$

Consider the second equation. To find the opposite of x+2, find the opposite of each term.

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

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

$$4\times \frac{y+1}{2}-4x-8+\frac{x-7}{3}+\frac{y+2}{2}=x$$

Cancel out 2, the greatest common factor in 4 and 2.

$$2\left(y+1\right)-4x-8+\frac{x-7}{3}+\frac{y+2}{2}=x$$

Use the distributive property to multiply 2 by y+1.

$$2y+2-4x-8+\frac{x-7}{3}+\frac{y+2}{2}=x$$

Subtract 8 from 2 to get -6.

$$2y-6-4x+\frac{x-7}{3}+\frac{y+2}{2}=x$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{x-7}{3} times \frac{2}{2}. Multiply \frac{y+2}{2} times \frac{3}{3}.

$$2y-6-4x+\frac{2\left(x-7\right)}{6}+\frac{3\left(y+2\right)}{6}=x$$

Since \frac{2\left(x-7\right)}{6} and \frac{3\left(y+2\right)}{6} have the same denominator, add them by adding their numerators.

$$2y-6-4x+\frac{2\left(x-7\right)+3\left(y+2\right)}{6}=x$$

Do the multiplications in 2\left(x-7\right)+3\left(y+2\right).

$$2y-6-4x+\frac{2x-14+3y+6}{6}=x$$

Combine like terms in 2x-14+3y+6.

$$2y-6-4x+\frac{2x-8+3y}{6}=x$$

Divide each term of 2x-8+3y by 6 to get \frac{1}{3}x-\frac{4}{3}+\frac{1}{2}y.

$$2y-6-4x+\frac{1}{3}x-\frac{4}{3}+\frac{1}{2}y=x$$

Combine -4x and \frac{1}{3}x to get -\frac{11}{3}x.

$$2y-6-\frac{11}{3}x-\frac{4}{3}+\frac{1}{2}y=x$$

Subtract \frac{4}{3} from -6 to get -\frac{22}{3}.

$$2y-\frac{22}{3}-\frac{11}{3}x+\frac{1}{2}y=x$$

Combine 2y and \frac{1}{2}y to get \frac{5}{2}y.

$$\frac{5}{2}y-\frac{22}{3}-\frac{11}{3}x=x$$

Subtract x from both sides.

$$\frac{5}{2}y-\frac{22}{3}-\frac{11}{3}x-x=0$$

Combine -\frac{11}{3}x and -x to get -\frac{14}{3}x.

$$\frac{5}{2}y-\frac{22}{3}-\frac{14}{3}x=0$$

Add \frac{22}{3} to both sides. Anything plus zero gives itself.

$$\frac{5}{2}y-\frac{14}{3}x=\frac{22}{3}$$

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

$$3x-2y=-\frac{11}{2},-\frac{14}{3}x+\frac{5}{2}y=\frac{22}{3}$$

Write the equations in matrix form.

$$\left(\begin{matrix}3&-2\\-\frac{14}{3}&\frac{5}{2}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{11}{2}\\\frac{22}{3}\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}3&-2\\-\frac{14}{3}&\frac{5}{2}\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-2\\-\frac{14}{3}&\frac{5}{2}\end{matrix}\right))\left(\begin{matrix}3&-2\\-\frac{14}{3}&\frac{5}{2}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-\frac{14}{3}&\frac{5}{2}\end{matrix}\right))\left(\begin{matrix}-\frac{11}{2}\\\frac{22}{3}\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-\frac{14}{3}&\frac{5}{2}\end{matrix}\right))\left(\begin{matrix}-\frac{11}{2}\\\frac{22}{3}\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-\frac{14}{3}&\frac{5}{2}\end{matrix}\right))\left(\begin{matrix}-\frac{11}{2}\\\frac{22}{3}\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{5}{2}}{3\times \frac{5}{2}-\left(-2\left(-\frac{14}{3}\right)\right)}&-\frac{-2}{3\times \frac{5}{2}-\left(-2\left(-\frac{14}{3}\right)\right)}\\-\frac{-\frac{14}{3}}{3\times \frac{5}{2}-\left(-2\left(-\frac{14}{3}\right)\right)}&\frac{3}{3\times \frac{5}{2}-\left(-2\left(-\frac{14}{3}\right)\right)}\end{matrix}\right)\left(\begin{matrix}-\frac{11}{2}\\\frac{22}{3}\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{15}{11}&-\frac{12}{11}\\-\frac{28}{11}&-\frac{18}{11}\end{matrix}\right)\left(\begin{matrix}-\frac{11}{2}\\\frac{22}{3}\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{15}{11}\left(-\frac{11}{2}\right)-\frac{12}{11}\times \frac{22}{3}\\-\frac{28}{11}\left(-\frac{11}{2}\right)-\frac{18}{11}\times \frac{22}{3}\end{matrix}\right)$$

Do the arithmetic.

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

Extract the matrix elements x and y.

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

Steps Using Elimination

Consider the first equation. Use the distributive property to multiply y by y-2x.

$$y^{2}-2yx+\left(x-y\right)\left(x+y\right)+3x-2y\left(1-x\right)=-\frac{11}{2}+x^{2}$$

Consider \left(x-y\right)\left(x+y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$y^{2}-2yx+x^{2}-y^{2}+3x-2y\left(1-x\right)=-\frac{11}{2}+x^{2}$$

Combine y^{2} and -y^{2} to get 0.

$$-2yx+x^{2}+3x-2y\left(1-x\right)=-\frac{11}{2}+x^{2}$$

Subtract x^{2} from both sides.

$$-2yx+x^{2}+3x-2y\left(1-x\right)-x^{2}=-\frac{11}{2}$$

Use the distributive property to multiply -2y by 1-x.

$$-2yx+x^{2}+3x-2y+2yx-x^{2}=-\frac{11}{2}$$

Combine -2yx and 2yx to get 0.

$$x^{2}+3x-2y-x^{2}=-\frac{11}{2}$$

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

$$3x-2y=-\frac{11}{2}$$

Consider the second equation. To find the opposite of x+2, find the opposite of each term.

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

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

$$4\times \frac{y+1}{2}-4x-8+\frac{x-7}{3}+\frac{y+2}{2}=x$$

Cancel out 2, the greatest common factor in 4 and 2.

$$2\left(y+1\right)-4x-8+\frac{x-7}{3}+\frac{y+2}{2}=x$$

Use the distributive property to multiply 2 by y+1.

$$2y+2-4x-8+\frac{x-7}{3}+\frac{y+2}{2}=x$$

Subtract 8 from 2 to get -6.

$$2y-6-4x+\frac{x-7}{3}+\frac{y+2}{2}=x$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{x-7}{3} times \frac{2}{2}. Multiply \frac{y+2}{2} times \frac{3}{3}.

$$2y-6-4x+\frac{2\left(x-7\right)}{6}+\frac{3\left(y+2\right)}{6}=x$$

Since \frac{2\left(x-7\right)}{6} and \frac{3\left(y+2\right)}{6} have the same denominator, add them by adding their numerators.

$$2y-6-4x+\frac{2\left(x-7\right)+3\left(y+2\right)}{6}=x$$

Do the multiplications in 2\left(x-7\right)+3\left(y+2\right).

$$2y-6-4x+\frac{2x-14+3y+6}{6}=x$$

Combine like terms in 2x-14+3y+6.

$$2y-6-4x+\frac{2x-8+3y}{6}=x$$

Divide each term of 2x-8+3y by 6 to get \frac{1}{3}x-\frac{4}{3}+\frac{1}{2}y.

$$2y-6-4x+\frac{1}{3}x-\frac{4}{3}+\frac{1}{2}y=x$$

Combine -4x and \frac{1}{3}x to get -\frac{11}{3}x.

$$2y-6-\frac{11}{3}x-\frac{4}{3}+\frac{1}{2}y=x$$

Subtract \frac{4}{3} from -6 to get -\frac{22}{3}.

$$2y-\frac{22}{3}-\frac{11}{3}x+\frac{1}{2}y=x$$

Combine 2y and \frac{1}{2}y to get \frac{5}{2}y.

$$\frac{5}{2}y-\frac{22}{3}-\frac{11}{3}x=x$$

Subtract x from both sides.

$$\frac{5}{2}y-\frac{22}{3}-\frac{11}{3}x-x=0$$

Combine -\frac{11}{3}x and -x to get -\frac{14}{3}x.

$$\frac{5}{2}y-\frac{22}{3}-\frac{14}{3}x=0$$

Add \frac{22}{3} to both sides. Anything plus zero gives itself.

$$\frac{5}{2}y-\frac{14}{3}x=\frac{22}{3}$$

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.

$$3x-2y=-\frac{11}{2},-\frac{14}{3}x+\frac{5}{2}y=\frac{22}{3}$$

To make 3x and -\frac{14x}{3} equal, multiply all terms on each side of the first equation by -\frac{14}{3} and all terms on each side of the second by 3.

$$-\frac{14}{3}\times 3x-\frac{14}{3}\left(-2\right)y=-\frac{14}{3}\left(-\frac{11}{2}\right),3\left(-\frac{14}{3}\right)x+3\times \frac{5}{2}y=3\times \frac{22}{3}$$

Simplify.

$$-14x+\frac{28}{3}y=\frac{77}{3},-14x+\frac{15}{2}y=22$$

Subtract -14x+\frac{15}{2}y=22 from -14x+\frac{28}{3}y=\frac{77}{3} by subtracting like terms on each side of the equal sign.

$$-14x+14x+\frac{28}{3}y-\frac{15}{2}y=\frac{77}{3}-22$$

Add -14x to 14x. Terms -14x and 14x cancel out, leaving an equation with only one variable that can be solved.

$$\frac{28}{3}y-\frac{15}{2}y=\frac{77}{3}-22$$

Add \frac{28y}{3} to -\frac{15y}{2}.

$$\frac{11}{6}y=\frac{77}{3}-22$$

Add \frac{77}{3} to -22.

$$\frac{11}{6}y=\frac{11}{3}$$

Divide both sides of the equation by \frac{11}{6}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=2$$

Substitute 2 for y in -\frac{14}{3}x+\frac{5}{2}y=\frac{22}{3}. Because the resulting equation contains only one variable, you can solve for x directly.

$$-\frac{14}{3}x+\frac{5}{2}\times 2=\frac{22}{3}$$

Multiply \frac{5}{2} times 2.

$$-\frac{14}{3}x+5=\frac{22}{3}$$

Subtract 5 from both sides of the equation.

$$-\frac{14}{3}x=\frac{7}{3}$$

Divide both sides of the equation by -\frac{14}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

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

The system is now solved.

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