Solve {l}{2x-1/2-2x+y-2/3=x+2/6}{x-y-1/2-frac{2x-y-3}{4}=frac{4x-y}{12}}

$$\left. \begin{array} { l } { \frac { 2 x - 1 } { 2 } - \frac { 2 x + y - 2 } { 3 } = \frac { x + 2 } { 6 } } \\ { \frac { x - y - 1 } { 2 } - \frac { 2 x - y - 3 } { 4 } = \frac { 4 x - y } { 12 } } \end{array} \right.$$

Solve for x, y

$$x=\frac{4}{5}=0.8$$

$$y=-\frac{1}{10}=-0.1$$

Steps Using Substitution

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

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

Use the distributive property to multiply 3 by 2x-1.

$$6x-3-2\left(2x+y-2\right)=x+2$$

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

$$6x-3-4x-2y+4=x+2$$

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

$$2x-3-2y+4=x+2$$

Add -3 and 4 to get 1.

$$2x+1-2y=x+2$$

Subtract x from both sides.

$$2x+1-2y-x=2$$

Combine 2x and -x to get x.

$$x+1-2y=2$$

Subtract 1 from both sides.

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

Subtract 1 from 2 to get 1.

$$x-2y=1$$

Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 2,4,12.

$$6\left(x-y-1\right)-3\left(2x-y-3\right)=4x-y$$

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

$$6x-6y-6-3\left(2x-y-3\right)=4x-y$$

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

$$6x-6y-6-6x+3y+9=4x-y$$

Combine 6x and -6x to get 0.

$$-6y-6+3y+9=4x-y$$

Combine -6y and 3y to get -3y.

$$-3y-6+9=4x-y$$

Add -6 and 9 to get 3.

$$-3y+3=4x-y$$

Subtract 4x from both sides.

$$-3y+3-4x=-y$$

Add y to both sides.

$$-3y+3-4x+y=0$$

Combine -3y and y to get -2y.

$$-2y+3-4x=0$$

Subtract 3 from both sides. Anything subtracted from zero gives its negation.

$$-2y-4x=-3$$

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.

$$x-2y=1,-4x-2y=-3$$

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

$$x-2y=1$$

Add 2y to both sides of the equation.

$$x=2y+1$$

Substitute 2y+1 for x in the other equation, -4x-2y=-3.

$$-4\left(2y+1\right)-2y=-3$$

Multiply -4 times 2y+1.

$$-8y-4-2y=-3$$

Add -8y to -2y.

$$-10y-4=-3$$

Add 4 to both sides of the equation.

$$-10y=1$$

Divide both sides by -10.

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

Substitute -\frac{1}{10} for y in x=2y+1. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=2\left(-\frac{1}{10}\right)+1$$

Multiply 2 times -\frac{1}{10}.

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

Add 1 to -\frac{1}{5}.

$$x=\frac{4}{5}$$

The system is now solved.

$$x=\frac{4}{5},y=-\frac{1}{10}$$

Steps Using Matrices

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

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

Use the distributive property to multiply 3 by 2x-1.

$$6x-3-2\left(2x+y-2\right)=x+2$$

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

$$6x-3-4x-2y+4=x+2$$

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

$$2x-3-2y+4=x+2$$

Add -3 and 4 to get 1.

$$2x+1-2y=x+2$$

Subtract x from both sides.

$$2x+1-2y-x=2$$

Combine 2x and -x to get x.

$$x+1-2y=2$$

Subtract 1 from both sides.

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

Subtract 1 from 2 to get 1.

$$x-2y=1$$

Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 2,4,12.

$$6\left(x-y-1\right)-3\left(2x-y-3\right)=4x-y$$

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

$$6x-6y-6-3\left(2x-y-3\right)=4x-y$$

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

$$6x-6y-6-6x+3y+9=4x-y$$

Combine 6x and -6x to get 0.

$$-6y-6+3y+9=4x-y$$

Combine -6y and 3y to get -3y.

$$-3y-6+9=4x-y$$

Add -6 and 9 to get 3.

$$-3y+3=4x-y$$

Subtract 4x from both sides.

$$-3y+3-4x=-y$$

Add y to both sides.

$$-3y+3-4x+y=0$$

Combine -3y and y to get -2y.

$$-2y+3-4x=0$$

Subtract 3 from both sides. Anything subtracted from zero gives its negation.

$$-2y-4x=-3$$

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

$$x-2y=1,-4x-2y=-3$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&-2\\-4&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\-3\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-2\\-4&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-2\\-4&-2\end{matrix}\right))\left(\begin{matrix}1&-2\\-4&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\-4&-2\end{matrix}\right))\left(\begin{matrix}1\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\-4&-2\end{matrix}\right))\left(\begin{matrix}1\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\-4&-2\end{matrix}\right))\left(\begin{matrix}1\\-3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-\left(-2\left(-4\right)\right)}&-\frac{-2}{-2-\left(-2\left(-4\right)\right)}\\-\frac{-4}{-2-\left(-2\left(-4\right)\right)}&\frac{1}{-2-\left(-2\left(-4\right)\right)}\end{matrix}\right)\left(\begin{matrix}1\\-3\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&-\frac{1}{5}\\-\frac{2}{5}&-\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}1\\-3\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}-\frac{1}{5}\left(-3\right)\\-\frac{2}{5}-\frac{1}{10}\left(-3\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{5}\\-\frac{1}{10}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{4}{5},y=-\frac{1}{10}$$

Steps Using Elimination

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

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

Use the distributive property to multiply 3 by 2x-1.

$$6x-3-2\left(2x+y-2\right)=x+2$$

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

$$6x-3-4x-2y+4=x+2$$

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

$$2x-3-2y+4=x+2$$

Add -3 and 4 to get 1.

$$2x+1-2y=x+2$$

Subtract x from both sides.

$$2x+1-2y-x=2$$

Combine 2x and -x to get x.

$$x+1-2y=2$$

Subtract 1 from both sides.

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

Subtract 1 from 2 to get 1.

$$x-2y=1$$

Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 2,4,12.

$$6\left(x-y-1\right)-3\left(2x-y-3\right)=4x-y$$

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

$$6x-6y-6-3\left(2x-y-3\right)=4x-y$$

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

$$6x-6y-6-6x+3y+9=4x-y$$

Combine 6x and -6x to get 0.

$$-6y-6+3y+9=4x-y$$

Combine -6y and 3y to get -3y.

$$-3y-6+9=4x-y$$

Add -6 and 9 to get 3.

$$-3y+3=4x-y$$

Subtract 4x from both sides.

$$-3y+3-4x=-y$$

Add y to both sides.

$$-3y+3-4x+y=0$$

Combine -3y and y to get -2y.

$$-2y+3-4x=0$$

Subtract 3 from both sides. Anything subtracted from zero gives its negation.

$$-2y-4x=-3$$

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.

$$x-2y=1,-4x-2y=-3$$

Subtract -4x-2y=-3 from x-2y=1 by subtracting like terms on each side of the equal sign.

$$x+4x-2y+2y=1+3$$

Add -2y to 2y. Terms -2y and 2y cancel out, leaving an equation with only one variable that can be solved.

$$x+4x=1+3$$

Add x to 4x.

$$5x=1+3$$

Add 1 to 3.

$$5x=4$$

Divide both sides by 5.

$$x=\frac{4}{5}$$

Substitute \frac{4}{5} for x in -4x-2y=-3. Because the resulting equation contains only one variable, you can solve for y directly.

$$-4\times \frac{4}{5}-2y=-3$$

Multiply -4 times \frac{4}{5}.

$$-\frac{16}{5}-2y=-3$$

Add \frac{16}{5} to both sides of the equation.

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

Divide both sides by -2.

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

The system is now solved.

$$x=\frac{4}{5},y=-\frac{1}{10}$$