Solve {l}{y=21x+180}{y=27x}

$$\left\{ \begin{array} { l } { y = 21 x + 180 } \\ { y = 27 x } \end{array} \right.$$

Solve for y, x

$$x=30$$

$$y=810$$

Steps Using Substitution

Consider the first equation. Subtract 21x from both sides.

$$y-21x=180$$

Consider the second equation. Subtract 27x from both sides.

$$y-27x=0$$

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.

$$y-21x=180,y-27x=0$$

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

$$y-21x=180$$

Add 21x to both sides of the equation.

$$y=21x+180$$

Substitute 21x+180 for y in the other equation, y-27x=0.

$$21x+180-27x=0$$

Add 21x to -27x.

$$-6x+180=0$$

Subtract 180 from both sides of the equation.

$$-6x=-180$$

Divide both sides by -6.

$$x=30$$

Substitute 30 for x in y=21x+180. Because the resulting equation contains only one variable, you can solve for y directly.

$$y=21\times 30+180$$

Multiply 21 times 30.

$$y=630+180$$

Add 180 to 630.

$$y=810$$

The system is now solved.

$$y=810,x=30$$

Steps Using Matrices

Consider the first equation. Subtract 21x from both sides.

$$y-21x=180$$

Consider the second equation. Subtract 27x from both sides.

$$y-27x=0$$

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

$$y-21x=180,y-27x=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&-21\\1&-27\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}180\\0\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-21\\1&-27\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-21\\1&-27\end{matrix}\right))\left(\begin{matrix}1&-21\\1&-27\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-21\\1&-27\end{matrix}\right))\left(\begin{matrix}180\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-21\\1&-27\end{matrix}\right))\left(\begin{matrix}180\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-21\\1&-27\end{matrix}\right))\left(\begin{matrix}180\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{27}{-27-\left(-21\right)}&-\frac{-21}{-27-\left(-21\right)}\\-\frac{1}{-27-\left(-21\right)}&\frac{1}{-27-\left(-21\right)}\end{matrix}\right)\left(\begin{matrix}180\\0\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{9}{2}&-\frac{7}{2}\\\frac{1}{6}&-\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}180\\0\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{9}{2}\times 180\\\frac{1}{6}\times 180\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}810\\30\end{matrix}\right)$$

Extract the matrix elements y and x.

$$y=810,x=30$$

Steps Using Elimination

Consider the first equation. Subtract 21x from both sides.

$$y-21x=180$$

Consider the second equation. Subtract 27x from both sides.

$$y-27x=0$$

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.

$$y-21x=180,y-27x=0$$

Subtract y-27x=0 from y-21x=180 by subtracting like terms on each side of the equal sign.

$$y-y-21x+27x=180$$

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

$$-21x+27x=180$$

Add -21x to 27x.

$$6x=180$$

Divide both sides by 6.

$$x=30$$

Substitute 30 for x in y-27x=0. Because the resulting equation contains only one variable, you can solve for y directly.

$$y-27\times 30=0$$

Multiply -27 times 30.

$$y-810=0$$

Add 810 to both sides of the equation.

$$y=810$$

The system is now solved.

$$y=810,x=30$$