Solve {l}{y=2x}{(10y+x)-(10x+y)=36}

$$\left\{ \begin{array} { l } { y = 2 x } \\ { ( 10 y + x ) - ( 10 x + y ) = 36 } \end{array} \right.$$

Solve for y, x

$$x=4$$

$$y=8$$

Steps Using Substitution

Consider the first equation. Subtract 2x from both sides.

$$y-2x=0$$

Consider the second equation. To find the opposite of 10x+y, find the opposite of each term.

$$10y+x-10x-y=36$$

Combine x and -10x to get -9x.

$$10y-9x-y=36$$

Combine 10y and -y to get 9y.

$$9y-9x=36$$

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.

$$y-2x=0,9y-9x=36$$

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

$$y-2x=0$$

Add 2x to both sides of the equation.

$$y=2x$$

Substitute 2x for y in the other equation, 9y-9x=36.

$$9\times 2x-9x=36$$

Multiply 9 times 2x.

$$18x-9x=36$$

Add 18x to -9x.

$$9x=36$$

Divide both sides by 9.

$$x=4$$

Substitute 4 for x in y=2x. Because the resulting equation contains only one variable, you can solve for y directly.

$$y=2\times 4$$

Multiply 2 times 4.

$$y=8$$

The system is now solved.

$$y=8,x=4$$

Steps Using Matrices

Consider the first equation. Subtract 2x from both sides.

$$y-2x=0$$

Consider the second equation. To find the opposite of 10x+y, find the opposite of each term.

$$10y+x-10x-y=36$$

Combine x and -10x to get -9x.

$$10y-9x-y=36$$

Combine 10y and -y to get 9y.

$$9y-9x=36$$

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

$$y-2x=0,9y-9x=36$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&-2\\9&-9\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\36\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-2\\9&-9\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-2\\9&-9\end{matrix}\right))\left(\begin{matrix}1&-2\\9&-9\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\9&-9\end{matrix}\right))\left(\begin{matrix}0\\36\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\9&-9\end{matrix}\right))\left(\begin{matrix}0\\36\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\9&-9\end{matrix}\right))\left(\begin{matrix}0\\36\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{-9-\left(-2\times 9\right)}&-\frac{-2}{-9-\left(-2\times 9\right)}\\-\frac{9}{-9-\left(-2\times 9\right)}&\frac{1}{-9-\left(-2\times 9\right)}\end{matrix}\right)\left(\begin{matrix}0\\36\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-1&\frac{2}{9}\\-1&\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}0\\36\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{9}\times 36\\\frac{1}{9}\times 36\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}8\\4\end{matrix}\right)$$

Extract the matrix elements y and x.

$$y=8,x=4$$

Steps Using Elimination

Consider the first equation. Subtract 2x from both sides.

$$y-2x=0$$

Consider the second equation. To find the opposite of 10x+y, find the opposite of each term.

$$10y+x-10x-y=36$$

Combine x and -10x to get -9x.

$$10y-9x-y=36$$

Combine 10y and -y to get 9y.

$$9y-9x=36$$

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.

$$y-2x=0,9y-9x=36$$

To make y and 9y equal, multiply all terms on each side of the first equation by 9 and all terms on each side of the second by 1.

$$9y+9\left(-2\right)x=0,9y-9x=36$$

Simplify.

$$9y-18x=0,9y-9x=36$$

Subtract 9y-9x=36 from 9y-18x=0 by subtracting like terms on each side of the equal sign.

$$9y-9y-18x+9x=-36$$

Add 9y to -9y. Terms 9y and -9y cancel out, leaving an equation with only one variable that can be solved.

$$-18x+9x=-36$$

Add -18x to 9x.

$$-9x=-36$$

Divide both sides by -9.

$$x=4$$

Substitute 4 for x in 9y-9x=36. Because the resulting equation contains only one variable, you can solve for y directly.

$$9y-9\times 4=36$$

Multiply -9 times 4.

$$9y-36=36$$

Add 36 to both sides of the equation.

$$9y=72$$

Divide both sides by 9.

$$y=8$$

The system is now solved.

$$y=8,x=4$$