Solve {l}{13x=y}{65x+2y=13}

$$\left\{ \begin{array} { l } { 13 x = y } \\ { 65 x + 2 y = 13 } \end{array} \right.$$

Solve for x, y

$$x=\frac{1}{7}\approx 0.142857143$$

$$y = \frac{13}{7} = 1\frac{6}{7} \approx 1.857142857$$

Steps Using Substitution

Consider the first equation. Subtract y from both sides.

$$13x-y=0$$

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.

$$13x-y=0,65x+2y=13$$

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

$$13x-y=0$$

Add y to both sides of the equation.

$$13x=y$$

Divide both sides by 13.

$$x=\frac{1}{13}y$$

Substitute \frac{y}{13} for x in the other equation, 65x+2y=13.

$$65\times \frac{1}{13}y+2y=13$$

Multiply 65 times \frac{y}{13}.

$$5y+2y=13$$

Add 5y to 2y.

$$7y=13$$

Divide both sides by 7.

$$y=\frac{13}{7}$$

Substitute \frac{13}{7} for y in x=\frac{1}{13}y. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{1}{13}\times \frac{13}{7}$$

Multiply \frac{1}{13} times \frac{13}{7} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{1}{7}$$

The system is now solved.

$$x=\frac{1}{7},y=\frac{13}{7}$$

Steps Using Matrices

Consider the first equation. Subtract y from both sides.

$$13x-y=0$$

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

$$13x-y=0,65x+2y=13$$

Write the equations in matrix form.

$$\left(\begin{matrix}13&-1\\65&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\13\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}13&-1\\65&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}13&-1\\65&2\end{matrix}\right))\left(\begin{matrix}13&-1\\65&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}13&-1\\65&2\end{matrix}\right))\left(\begin{matrix}0\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}13&-1\\65&2\end{matrix}\right))\left(\begin{matrix}0\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}13&-1\\65&2\end{matrix}\right))\left(\begin{matrix}0\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{13\times 2-\left(-65\right)}&-\frac{-1}{13\times 2-\left(-65\right)}\\-\frac{65}{13\times 2-\left(-65\right)}&\frac{13}{13\times 2-\left(-65\right)}\end{matrix}\right)\left(\begin{matrix}0\\13\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{91}&\frac{1}{91}\\-\frac{5}{7}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}0\\13\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{91}\times 13\\\frac{1}{7}\times 13\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}\\\frac{13}{7}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{1}{7},y=\frac{13}{7}$$

Steps Using Elimination

Consider the first equation. Subtract y from both sides.

$$13x-y=0$$

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.

$$13x-y=0,65x+2y=13$$

To make 13x and 65x equal, multiply all terms on each side of the first equation by 65 and all terms on each side of the second by 13.

$$65\times 13x+65\left(-1\right)y=0,13\times 65x+13\times 2y=13\times 13$$

Simplify.

$$845x-65y=0,845x+26y=169$$

Subtract 845x+26y=169 from 845x-65y=0 by subtracting like terms on each side of the equal sign.

$$845x-845x-65y-26y=-169$$

Add 845x to -845x. Terms 845x and -845x cancel out, leaving an equation with only one variable that can be solved.

$$-65y-26y=-169$$

Add -65y to -26y.

$$-91y=-169$$

Divide both sides by -91.

$$y=\frac{13}{7}$$

Substitute \frac{13}{7} for y in 65x+2y=13. Because the resulting equation contains only one variable, you can solve for x directly.

$$65x+2\times \frac{13}{7}=13$$

Multiply 2 times \frac{13}{7}.

$$65x+\frac{26}{7}=13$$

Subtract \frac{26}{7} from both sides of the equation.

$$65x=\frac{65}{7}$$

Divide both sides by 65.

$$x=\frac{1}{7}$$

The system is now solved.

$$x=\frac{1}{7},y=\frac{13}{7}$$