Solve {l}{23x+7y=13}{101x+73y=1}

$$\left. \begin{array} { l } { 23 x + 7 y = 13 } \\ { 101 x + 73 y = 1 } \end{array} \right.$$

Solve for x, y

$$x=\frac{157}{162}\approx 0.969135802$$

$$y = -\frac{215}{162} = -1\frac{53}{162} \approx -1.327160494$$

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.

$$23x+7y=13,101x+73y=1$$

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

$$23x+7y=13$$

Subtract 7y from both sides of the equation.

$$23x=-7y+13$$

Divide both sides by 23.

$$x=\frac{1}{23}\left(-7y+13\right)$$

Multiply \frac{1}{23} times -7y+13.

$$x=-\frac{7}{23}y+\frac{13}{23}$$

Substitute \frac{-7y+13}{23} for x in the other equation, 101x+73y=1.

$$101\left(-\frac{7}{23}y+\frac{13}{23}\right)+73y=1$$

Multiply 101 times \frac{-7y+13}{23}.

$$-\frac{707}{23}y+\frac{1313}{23}+73y=1$$

Add -\frac{707y}{23} to 73y.

$$\frac{972}{23}y+\frac{1313}{23}=1$$

Subtract \frac{1313}{23} from both sides of the equation.

$$\frac{972}{23}y=-\frac{1290}{23}$$

Divide both sides of the equation by \frac{972}{23}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-\frac{215}{162}$$

Substitute -\frac{215}{162} for y in x=-\frac{7}{23}y+\frac{13}{23}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{7}{23}\left(-\frac{215}{162}\right)+\frac{13}{23}$$

Multiply -\frac{7}{23} times -\frac{215}{162} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{1505}{3726}+\frac{13}{23}$$

Add \frac{13}{23} to \frac{1505}{3726} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{157}{162}$$

The system is now solved.

$$x=\frac{157}{162},y=-\frac{215}{162}$$

Steps Using Matrices

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

$$23x+7y=13,101x+73y=1$$

Write the equations in matrix form.

$$\left(\begin{matrix}23&7\\101&73\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}13\\1\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}23&7\\101&73\end{matrix}\right).

$$inverse(\left(\begin{matrix}23&7\\101&73\end{matrix}\right))\left(\begin{matrix}23&7\\101&73\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}23&7\\101&73\end{matrix}\right))\left(\begin{matrix}13\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}23&7\\101&73\end{matrix}\right))\left(\begin{matrix}13\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}23&7\\101&73\end{matrix}\right))\left(\begin{matrix}13\\1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{73}{23\times 73-7\times 101}&-\frac{7}{23\times 73-7\times 101}\\-\frac{101}{23\times 73-7\times 101}&\frac{23}{23\times 73-7\times 101}\end{matrix}\right)\left(\begin{matrix}13\\1\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{73}{972}&-\frac{7}{972}\\-\frac{101}{972}&\frac{23}{972}\end{matrix}\right)\left(\begin{matrix}13\\1\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{73}{972}\times 13-\frac{7}{972}\\-\frac{101}{972}\times 13+\frac{23}{972}\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{157}{162}\\-\frac{215}{162}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{157}{162},y=-\frac{215}{162}$$

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.

$$23x+7y=13,101x+73y=1$$

To make 23x and 101x equal, multiply all terms on each side of the first equation by 101 and all terms on each side of the second by 23.

$$101\times 23x+101\times 7y=101\times 13,23\times 101x+23\times 73y=23$$

Simplify.

$$2323x+707y=1313,2323x+1679y=23$$

Subtract 2323x+1679y=23 from 2323x+707y=1313 by subtracting like terms on each side of the equal sign.

$$2323x-2323x+707y-1679y=1313-23$$

Add 2323x to -2323x. Terms 2323x and -2323x cancel out, leaving an equation with only one variable that can be solved.

$$707y-1679y=1313-23$$

Add 707y to -1679y.

$$-972y=1313-23$$

Add 1313 to -23.

$$-972y=1290$$

Divide both sides by -972.

$$y=-\frac{215}{162}$$

Substitute -\frac{215}{162} for y in 101x+73y=1. Because the resulting equation contains only one variable, you can solve for x directly.

$$101x+73\left(-\frac{215}{162}\right)=1$$

Multiply 73 times -\frac{215}{162}.

$$101x-\frac{15695}{162}=1$$

Add \frac{15695}{162} to both sides of the equation.

$$101x=\frac{15857}{162}$$

Divide both sides by 101.

$$x=\frac{157}{162}$$

The system is now solved.

$$x=\frac{157}{162},y=-\frac{215}{162}$$