Solve {l}{y+2x=21}{10x+y=10y+x+27}

$$\left\{ \begin{array} { l } { y + 2 x = 21 } \\ { 10 x + y = 10 y + x + 27 } \end{array} \right.$$

Solve for y, x

$$x=8$$

$$y=5$$

Steps Using Substitution

Consider the second equation. Subtract 10y from both sides.

$$10x+y-10y=x+27$$

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

$$10x-9y=x+27$$

Subtract x from both sides.

$$10x-9y-x=27$$

Combine 10x and -x to get 9x.

$$9x-9y=27$$

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.

$$y+2x=21,-9y+9x=27$$

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

$$y+2x=21$$

Subtract 2x from both sides of the equation.

$$y=-2x+21$$

Substitute -2x+21 for y in the other equation, -9y+9x=27.

$$-9\left(-2x+21\right)+9x=27$$

Multiply -9 times -2x+21.

$$18x-189+9x=27$$

Add 18x to 9x.

$$27x-189=27$$

Add 189 to both sides of the equation.

$$27x=216$$

Divide both sides by 27.

$$x=8$$

Substitute 8 for x in y=-2x+21. Because the resulting equation contains only one variable, you can solve for y directly.

$$y=-2\times 8+21$$

Multiply -2 times 8.

$$y=-16+21$$

Add 21 to -16.

$$y=5$$

The system is now solved.

$$y=5,x=8$$

Steps Using Matrices

Consider the second equation. Subtract 10y from both sides.

$$10x+y-10y=x+27$$

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

$$10x-9y=x+27$$

Subtract x from both sides.

$$10x-9y-x=27$$

Combine 10x and -x to get 9x.

$$9x-9y=27$$

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

$$y+2x=21,-9y+9x=27$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&2\\-9&9\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}21\\27\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&2\\-9&9\end{matrix}\right))\left(\begin{matrix}1&2\\-9&9\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-9&9\end{matrix}\right))\left(\begin{matrix}21\\27\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-9&9\end{matrix}\right))\left(\begin{matrix}21\\27\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-9&9\end{matrix}\right))\left(\begin{matrix}21\\27\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{9}{9-2\left(-9\right)}&-\frac{2}{9-2\left(-9\right)}\\-\frac{-9}{9-2\left(-9\right)}&\frac{1}{9-2\left(-9\right)}\end{matrix}\right)\left(\begin{matrix}21\\27\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&-\frac{2}{27}\\\frac{1}{3}&\frac{1}{27}\end{matrix}\right)\left(\begin{matrix}21\\27\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 21-\frac{2}{27}\times 27\\\frac{1}{3}\times 21+\frac{1}{27}\times 27\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}5\\8\end{matrix}\right)$$

Extract the matrix elements y and x.

$$y=5,x=8$$

Steps Using Elimination

Consider the second equation. Subtract 10y from both sides.

$$10x+y-10y=x+27$$

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

$$10x-9y=x+27$$

Subtract x from both sides.

$$10x-9y-x=27$$

Combine 10x and -x to get 9x.

$$9x-9y=27$$

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.

$$y+2x=21,-9y+9x=27$$

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.

$$-9y-9\times 2x=-9\times 21,-9y+9x=27$$

Simplify.

$$-9y-18x=-189,-9y+9x=27$$

Subtract -9y+9x=27 from -9y-18x=-189 by subtracting like terms on each side of the equal sign.

$$-9y+9y-18x-9x=-189-27$$

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

$$-18x-9x=-189-27$$

Add -18x to -9x.

$$-27x=-189-27$$

Add -189 to -27.

$$-27x=-216$$

Divide both sides by -27.

$$x=8$$

Substitute 8 for x in -9y+9x=27. Because the resulting equation contains only one variable, you can solve for y directly.

$$-9y+9\times 8=27$$

Multiply 9 times 8.

$$-9y+72=27$$

Subtract 72 from both sides of the equation.

$$-9y=-45$$

Divide both sides by -9.

$$y=5$$

The system is now solved.

$$y=5,x=8$$