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

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

Solve for x, y

$$x=-\frac{1}{22}\approx -0.045454545$$

$$y=-\frac{3}{11}\approx -0.272727273$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 2.

$$2\left(x-y\right)^{2}+2x+7y=2x\left(x-2y\right)+2y^{2}-2$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-y\right)^{2}.

$$2\left(x^{2}-2xy+y^{2}\right)+2x+7y=2x\left(x-2y\right)+2y^{2}-2$$

Use the distributive property to multiply 2 by x^{2}-2xy+y^{2}.

$$2x^{2}-4xy+2y^{2}+2x+7y=2x\left(x-2y\right)+2y^{2}-2$$

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

$$2x^{2}-4xy+2y^{2}+2x+7y=2x^{2}-4xy+2y^{2}-2$$

Subtract 2x^{2} from both sides.

$$2x^{2}-4xy+2y^{2}+2x+7y-2x^{2}=-4xy+2y^{2}-2$$

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

$$-4xy+2y^{2}+2x+7y=-4xy+2y^{2}-2$$

Add 4xy to both sides.

$$-4xy+2y^{2}+2x+7y+4xy=2y^{2}-2$$

Combine -4xy and 4xy to get 0.

$$2y^{2}+2x+7y=2y^{2}-2$$

Subtract 2y^{2} from both sides.

$$2y^{2}+2x+7y-2y^{2}=-2$$

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

$$2x+7y=-2$$

Consider the second equation. Multiply both sides of the equation by 2.

$$6x-\left(y-2\right)=2$$

To find the opposite of y-2, find the opposite of each term.

$$6x-y+2=2$$

Subtract 2 from both sides.

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

Subtract 2 from 2 to get 0.

$$6x-y=0$$

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.

$$2x+7y=-2,6x-y=0$$

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

$$2x+7y=-2$$

Subtract 7y from both sides of the equation.

$$2x=-7y-2$$

Divide both sides by 2.

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

Multiply \frac{1}{2} times -7y-2.

$$x=-\frac{7}{2}y-1$$

Substitute -\frac{7y}{2}-1 for x in the other equation, 6x-y=0.

$$6\left(-\frac{7}{2}y-1\right)-y=0$$

Multiply 6 times -\frac{7y}{2}-1.

$$-21y-6-y=0$$

Add -21y to -y.

$$-22y-6=0$$

Add 6 to both sides of the equation.

$$-22y=6$$

Divide both sides by -22.

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

Substitute -\frac{3}{11} for y in x=-\frac{7}{2}y-1. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{7}{2}\left(-\frac{3}{11}\right)-1$$

Multiply -\frac{7}{2} times -\frac{3}{11} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{21}{22}-1$$

Add -1 to \frac{21}{22}.

$$x=-\frac{1}{22}$$

The system is now solved.

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

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 2.

$$2\left(x-y\right)^{2}+2x+7y=2x\left(x-2y\right)+2y^{2}-2$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-y\right)^{2}.

$$2\left(x^{2}-2xy+y^{2}\right)+2x+7y=2x\left(x-2y\right)+2y^{2}-2$$

Use the distributive property to multiply 2 by x^{2}-2xy+y^{2}.

$$2x^{2}-4xy+2y^{2}+2x+7y=2x\left(x-2y\right)+2y^{2}-2$$

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

$$2x^{2}-4xy+2y^{2}+2x+7y=2x^{2}-4xy+2y^{2}-2$$

Subtract 2x^{2} from both sides.

$$2x^{2}-4xy+2y^{2}+2x+7y-2x^{2}=-4xy+2y^{2}-2$$

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

$$-4xy+2y^{2}+2x+7y=-4xy+2y^{2}-2$$

Add 4xy to both sides.

$$-4xy+2y^{2}+2x+7y+4xy=2y^{2}-2$$

Combine -4xy and 4xy to get 0.

$$2y^{2}+2x+7y=2y^{2}-2$$

Subtract 2y^{2} from both sides.

$$2y^{2}+2x+7y-2y^{2}=-2$$

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

$$2x+7y=-2$$

Consider the second equation. Multiply both sides of the equation by 2.

$$6x-\left(y-2\right)=2$$

To find the opposite of y-2, find the opposite of each term.

$$6x-y+2=2$$

Subtract 2 from both sides.

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

Subtract 2 from 2 to get 0.

$$6x-y=0$$

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

$$2x+7y=-2,6x-y=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}2&7\\6&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\\0\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&7\\6&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&7\\6&-1\end{matrix}\right))\left(\begin{matrix}2&7\\6&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&7\\6&-1\end{matrix}\right))\left(\begin{matrix}-2\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&7\\6&-1\end{matrix}\right))\left(\begin{matrix}-2\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&7\\6&-1\end{matrix}\right))\left(\begin{matrix}-2\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2\left(-1\right)-7\times 6}&-\frac{7}{2\left(-1\right)-7\times 6}\\-\frac{6}{2\left(-1\right)-7\times 6}&\frac{2}{2\left(-1\right)-7\times 6}\end{matrix}\right)\left(\begin{matrix}-2\\0\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{44}&\frac{7}{44}\\\frac{3}{22}&-\frac{1}{22}\end{matrix}\right)\left(\begin{matrix}-2\\0\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{44}\left(-2\right)\\\frac{3}{22}\left(-2\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{22}\\-\frac{3}{11}\end{matrix}\right)$$

Extract the matrix elements x and y.

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

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 2.

$$2\left(x-y\right)^{2}+2x+7y=2x\left(x-2y\right)+2y^{2}-2$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-y\right)^{2}.

$$2\left(x^{2}-2xy+y^{2}\right)+2x+7y=2x\left(x-2y\right)+2y^{2}-2$$

Use the distributive property to multiply 2 by x^{2}-2xy+y^{2}.

$$2x^{2}-4xy+2y^{2}+2x+7y=2x\left(x-2y\right)+2y^{2}-2$$

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

$$2x^{2}-4xy+2y^{2}+2x+7y=2x^{2}-4xy+2y^{2}-2$$

Subtract 2x^{2} from both sides.

$$2x^{2}-4xy+2y^{2}+2x+7y-2x^{2}=-4xy+2y^{2}-2$$

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

$$-4xy+2y^{2}+2x+7y=-4xy+2y^{2}-2$$

Add 4xy to both sides.

$$-4xy+2y^{2}+2x+7y+4xy=2y^{2}-2$$

Combine -4xy and 4xy to get 0.

$$2y^{2}+2x+7y=2y^{2}-2$$

Subtract 2y^{2} from both sides.

$$2y^{2}+2x+7y-2y^{2}=-2$$

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

$$2x+7y=-2$$

Consider the second equation. Multiply both sides of the equation by 2.

$$6x-\left(y-2\right)=2$$

To find the opposite of y-2, find the opposite of each term.

$$6x-y+2=2$$

Subtract 2 from both sides.

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

Subtract 2 from 2 to get 0.

$$6x-y=0$$

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.

$$2x+7y=-2,6x-y=0$$

To make 2x and 6x equal, multiply all terms on each side of the first equation by 6 and all terms on each side of the second by 2.

$$6\times 2x+6\times 7y=6\left(-2\right),2\times 6x+2\left(-1\right)y=0$$

Simplify.

$$12x+42y=-12,12x-2y=0$$

Subtract 12x-2y=0 from 12x+42y=-12 by subtracting like terms on each side of the equal sign.

$$12x-12x+42y+2y=-12$$

Add 12x to -12x. Terms 12x and -12x cancel out, leaving an equation with only one variable that can be solved.

$$42y+2y=-12$$

Add 42y to 2y.

$$44y=-12$$

Divide both sides by 44.

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

Substitute -\frac{3}{11} for y in 6x-y=0. Because the resulting equation contains only one variable, you can solve for x directly.

$$6x-\left(-\frac{3}{11}\right)=0$$

Subtract \frac{3}{11} from both sides of the equation.

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

Divide both sides by 6.

$$x=-\frac{1}{22}$$

The system is now solved.

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