Solve {l}{27673x-337y=38}{99887x+66y=887766}

$$\left. \begin{array} { l } { 27673 x - 337 y = 38 } \\ { 99887 x + 66 y = 887766 } \end{array} \right.$$

Solve for x, y

$$x = \frac{299179650}{35488337} = 8\frac{15272954}{35488337} \approx 8.430365447$$

$$y = \frac{24563352812}{35488337} = 692\frac{5423608}{35488337} \approx 692.152827899$$

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.

$$27673x-337y=38,99887x+66y=887766$$

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

$$27673x-337y=38$$

Add 337y to both sides of the equation.

$$27673x=337y+38$$

Divide both sides by 27673.

$$x=\frac{1}{27673}\left(337y+38\right)$$

Multiply \frac{1}{27673} times 337y+38.

$$x=\frac{337}{27673}y+\frac{38}{27673}$$

Substitute \frac{337y+38}{27673} for x in the other equation, 99887x+66y=887766.

$$99887\left(\frac{337}{27673}y+\frac{38}{27673}\right)+66y=887766$$

Multiply 99887 times \frac{337y+38}{27673}.

$$\frac{33661919}{27673}y+\frac{3795706}{27673}+66y=887766$$

Add \frac{33661919y}{27673} to 66y.

$$\frac{35488337}{27673}y+\frac{3795706}{27673}=887766$$

Subtract \frac{3795706}{27673} from both sides of the equation.

$$\frac{35488337}{27673}y=\frac{24563352812}{27673}$$

Divide both sides of the equation by \frac{35488337}{27673}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{24563352812}{35488337}$$

Substitute \frac{24563352812}{35488337} for y in x=\frac{337}{27673}y+\frac{38}{27673}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{337}{27673}\times \frac{24563352812}{35488337}+\frac{38}{27673}$$

Multiply \frac{337}{27673} times \frac{24563352812}{35488337} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{8277849897644}{982068749801}+\frac{38}{27673}$$

Add \frac{38}{27673} to \frac{8277849897644}{982068749801} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{299179650}{35488337}$$

The system is now solved.

$$x=\frac{299179650}{35488337},y=\frac{24563352812}{35488337}$$

Steps Using Matrices

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

$$27673x-337y=38,99887x+66y=887766$$

Write the equations in matrix form.

$$\left(\begin{matrix}27673&-337\\99887&66\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}38\\887766\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}27673&-337\\99887&66\end{matrix}\right).

$$inverse(\left(\begin{matrix}27673&-337\\99887&66\end{matrix}\right))\left(\begin{matrix}27673&-337\\99887&66\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}27673&-337\\99887&66\end{matrix}\right))\left(\begin{matrix}38\\887766\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}27673&-337\\99887&66\end{matrix}\right))\left(\begin{matrix}38\\887766\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}27673&-337\\99887&66\end{matrix}\right))\left(\begin{matrix}38\\887766\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{66}{27673\times 66-\left(-337\times 99887\right)}&-\frac{-337}{27673\times 66-\left(-337\times 99887\right)}\\-\frac{99887}{27673\times 66-\left(-337\times 99887\right)}&\frac{27673}{27673\times 66-\left(-337\times 99887\right)}\end{matrix}\right)\left(\begin{matrix}38\\887766\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{66}{35488337}&\frac{337}{35488337}\\-\frac{99887}{35488337}&\frac{27673}{35488337}\end{matrix}\right)\left(\begin{matrix}38\\887766\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{66}{35488337}\times 38+\frac{337}{35488337}\times 887766\\-\frac{99887}{35488337}\times 38+\frac{27673}{35488337}\times 887766\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{299179650}{35488337}\\\frac{24563352812}{35488337}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{299179650}{35488337},y=\frac{24563352812}{35488337}$$

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.

$$27673x-337y=38,99887x+66y=887766$$

To make 27673x and 99887x equal, multiply all terms on each side of the first equation by 99887 and all terms on each side of the second by 27673.

$$99887\times 27673x+99887\left(-337\right)y=99887\times 38,27673\times 99887x+27673\times 66y=27673\times 887766$$

Simplify.

$$2764172951x-33661919y=3795706,2764172951x+1826418y=24567148518$$

Subtract 2764172951x+1826418y=24567148518 from 2764172951x-33661919y=3795706 by subtracting like terms on each side of the equal sign.

$$2764172951x-2764172951x-33661919y-1826418y=3795706-24567148518$$

Add 2764172951x to -2764172951x. Terms 2764172951x and -2764172951x cancel out, leaving an equation with only one variable that can be solved.

$$-33661919y-1826418y=3795706-24567148518$$

Add -33661919y to -1826418y.

$$-35488337y=3795706-24567148518$$

Add 3795706 to -24567148518.

$$-35488337y=-24563352812$$

Divide both sides by -35488337.

$$y=\frac{24563352812}{35488337}$$

Substitute \frac{24563352812}{35488337} for y in 99887x+66y=887766. Because the resulting equation contains only one variable, you can solve for x directly.

$$99887x+66\times \frac{24563352812}{35488337}=887766$$

Multiply 66 times \frac{24563352812}{35488337}.

$$99887x+\frac{1621181285592}{35488337}=887766$$

Subtract \frac{1621181285592}{35488337} from both sides of the equation.

$$99887x=\frac{29884157699550}{35488337}$$

Divide both sides by 99887.

$$x=\frac{299179650}{35488337}$$

The system is now solved.

$$x=\frac{299179650}{35488337},y=\frac{24563352812}{35488337}$$