Solve {l}{3(x+y)-4(x-y)=1}{x+y/2-x-y/6=1}

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

Solve for x, y

$$x = \frac{19}{9} = 2\frac{1}{9} \approx 2.111111111$$

$$y=\frac{4}{9}\approx 0.444444444$$

Steps Using Substitution

Consider the first equation. Use the distributive property to multiply 3 by x+y.

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

Use the distributive property to multiply -4 by x-y.

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

Combine 3x and -4x to get -x.

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

Combine 3y and 4y to get 7y.

$$-x+7y=1$$

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

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

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

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

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

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

Combine 3x and -x to get 2x.

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

Combine 3y and y to get 4y.

$$2x+4y=6$$

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.

$$-x+7y=1,2x+4y=6$$

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

$$-x+7y=1$$

Subtract 7y from both sides of the equation.

$$-x=-7y+1$$

Divide both sides by -1.

$$x=-\left(-7y+1\right)$$

Multiply -1 times -7y+1.

$$x=7y-1$$

Substitute 7y-1 for x in the other equation, 2x+4y=6.

$$2\left(7y-1\right)+4y=6$$

Multiply 2 times 7y-1.

$$14y-2+4y=6$$

Add 14y to 4y.

$$18y-2=6$$

Add 2 to both sides of the equation.

$$18y=8$$

Divide both sides by 18.

$$y=\frac{4}{9}$$

Substitute \frac{4}{9} for y in x=7y-1. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=7\times \frac{4}{9}-1$$

Multiply 7 times \frac{4}{9}.

$$x=\frac{28}{9}-1$$

Add -1 to \frac{28}{9}.

$$x=\frac{19}{9}$$

The system is now solved.

$$x=\frac{19}{9},y=\frac{4}{9}$$

Steps Using Matrices

Consider the first equation. Use the distributive property to multiply 3 by x+y.

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

Use the distributive property to multiply -4 by x-y.

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

Combine 3x and -4x to get -x.

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

Combine 3y and 4y to get 7y.

$$-x+7y=1$$

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

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

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

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

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

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

Combine 3x and -x to get 2x.

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

Combine 3y and y to get 4y.

$$2x+4y=6$$

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

$$-x+7y=1,2x+4y=6$$

Write the equations in matrix form.

$$\left(\begin{matrix}-1&7\\2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\6\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}-1&7\\2&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}-1&7\\2&4\end{matrix}\right))\left(\begin{matrix}-1&7\\2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&7\\2&4\end{matrix}\right))\left(\begin{matrix}1\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&7\\2&4\end{matrix}\right))\left(\begin{matrix}1\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&7\\2&4\end{matrix}\right))\left(\begin{matrix}1\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{-4-7\times 2}&-\frac{7}{-4-7\times 2}\\-\frac{2}{-4-7\times 2}&-\frac{1}{-4-7\times 2}\end{matrix}\right)\left(\begin{matrix}1\\6\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{9}&\frac{7}{18}\\\frac{1}{9}&\frac{1}{18}\end{matrix}\right)\left(\begin{matrix}1\\6\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{9}+\frac{7}{18}\times 6\\\frac{1}{9}+\frac{1}{18}\times 6\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{19}{9}\\\frac{4}{9}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{19}{9},y=\frac{4}{9}$$

Steps Using Elimination

Consider the first equation. Use the distributive property to multiply 3 by x+y.

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

Use the distributive property to multiply -4 by x-y.

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

Combine 3x and -4x to get -x.

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

Combine 3y and 4y to get 7y.

$$-x+7y=1$$

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

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

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

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

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

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

Combine 3x and -x to get 2x.

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

Combine 3y and y to get 4y.

$$2x+4y=6$$

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.

$$-x+7y=1,2x+4y=6$$

To make -x and 2x equal, multiply all terms on each side of the first equation by 2 and all terms on each side of the second by -1.

$$2\left(-1\right)x+2\times 7y=2,-2x-4y=-6$$

Simplify.

$$-2x+14y=2,-2x-4y=-6$$

Subtract -2x-4y=-6 from -2x+14y=2 by subtracting like terms on each side of the equal sign.

$$-2x+2x+14y+4y=2+6$$

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

$$14y+4y=2+6$$

Add 14y to 4y.

$$18y=2+6$$

Add 2 to 6.

$$18y=8$$

Divide both sides by 18.

$$y=\frac{4}{9}$$

Substitute \frac{4}{9} for y in 2x+4y=6. Because the resulting equation contains only one variable, you can solve for x directly.

$$2x+4\times \frac{4}{9}=6$$

Multiply 4 times \frac{4}{9}.

$$2x+\frac{16}{9}=6$$

Subtract \frac{16}{9} from both sides of the equation.

$$2x=\frac{38}{9}$$

Divide both sides by 2.

$$x=\frac{19}{9}$$

The system is now solved.

$$x=\frac{19}{9},y=\frac{4}{9}$$