Solve {l}{16x+18y=16}{17x+y=20}

$$\left. \begin{array} { l } { 16 x + 18 y = 16 } \\ { 17 x + y = 20 } \end{array} \right.$$

Solve for x, y

$$x = \frac{172}{145} = 1\frac{27}{145} \approx 1.186206897$$

$$y=-\frac{24}{145}\approx -0.165517241$$

Steps Using Substitution

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.

$$16x+18y=16,17x+y=20$$

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

$$16x+18y=16$$

Subtract 18y from both sides of the equation.

$$16x=-18y+16$$

Divide both sides by 16.

$$x=\frac{1}{16}\left(-18y+16\right)$$

Multiply \frac{1}{16} times -18y+16.

$$x=-\frac{9}{8}y+1$$

Substitute -\frac{9y}{8}+1 for x in the other equation, 17x+y=20.

$$17\left(-\frac{9}{8}y+1\right)+y=20$$

Multiply 17 times -\frac{9y}{8}+1.

$$-\frac{153}{8}y+17+y=20$$

Add -\frac{153y}{8} to y.

$$-\frac{145}{8}y+17=20$$

Subtract 17 from both sides of the equation.

$$-\frac{145}{8}y=3$$

Divide both sides of the equation by -\frac{145}{8}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-\frac{24}{145}$$

Substitute -\frac{24}{145} for y in x=-\frac{9}{8}y+1. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{9}{8}\left(-\frac{24}{145}\right)+1$$

Multiply -\frac{9}{8} times -\frac{24}{145} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{27}{145}+1$$

Add 1 to \frac{27}{145}.

$$x=\frac{172}{145}$$

The system is now solved.

$$x=\frac{172}{145},y=-\frac{24}{145}$$

Steps Using Matrices

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

$$16x+18y=16,17x+y=20$$

Write the equations in matrix form.

$$\left(\begin{matrix}16&18\\17&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}16\\20\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}16&18\\17&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}16&18\\17&1\end{matrix}\right))\left(\begin{matrix}16&18\\17&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}16&18\\17&1\end{matrix}\right))\left(\begin{matrix}16\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}16&18\\17&1\end{matrix}\right))\left(\begin{matrix}16\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}16&18\\17&1\end{matrix}\right))\left(\begin{matrix}16\\20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{16-18\times 17}&-\frac{18}{16-18\times 17}\\-\frac{17}{16-18\times 17}&\frac{16}{16-18\times 17}\end{matrix}\right)\left(\begin{matrix}16\\20\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{290}&\frac{9}{145}\\\frac{17}{290}&-\frac{8}{145}\end{matrix}\right)\left(\begin{matrix}16\\20\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{290}\times 16+\frac{9}{145}\times 20\\\frac{17}{290}\times 16-\frac{8}{145}\times 20\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{172}{145}\\-\frac{24}{145}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{172}{145},y=-\frac{24}{145}$$

Steps Using Elimination

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.

$$16x+18y=16,17x+y=20$$

To make 16x and 17x equal, multiply all terms on each side of the first equation by 17 and all terms on each side of the second by 16.

$$17\times 16x+17\times 18y=17\times 16,16\times 17x+16y=16\times 20$$

Simplify.

$$272x+306y=272,272x+16y=320$$

Subtract 272x+16y=320 from 272x+306y=272 by subtracting like terms on each side of the equal sign.

$$272x-272x+306y-16y=272-320$$

Add 272x to -272x. Terms 272x and -272x cancel out, leaving an equation with only one variable that can be solved.

$$306y-16y=272-320$$

Add 306y to -16y.

$$290y=272-320$$

Add 272 to -320.

$$290y=-48$$

Divide both sides by 290.

$$y=-\frac{24}{145}$$

Substitute -\frac{24}{145} for y in 17x+y=20. Because the resulting equation contains only one variable, you can solve for x directly.

$$17x-\frac{24}{145}=20$$

Add \frac{24}{145} to both sides of the equation.

$$17x=\frac{2924}{145}$$

Divide both sides by 17.

$$x=\frac{172}{145}$$

The system is now solved.

$$x=\frac{172}{145},y=-\frac{24}{145}$$