Solve {l}{x/92+y=58}{x-y/79=57}

$$\left. \begin{array} { l } { \frac { x } { 92 } + y = 58 } \\ { x - \frac { y } { 79 } = 57 } \end{array} \right.$$

Solve for x, y

$$x = \frac{419612}{7269} = 57\frac{5279}{7269} \approx 57.726234695$$

$$y = \frac{417041}{7269} = 57\frac{2708}{7269} \approx 57.372540927$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 92.

$$x+92y=5336$$

Consider the second equation. Multiply both sides of the equation by 79.

$$79x-y=4503$$

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.

$$x+92y=5336,79x-y=4503$$

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

$$x+92y=5336$$

Subtract 92y from both sides of the equation.

$$x=-92y+5336$$

Substitute -92y+5336 for x in the other equation, 79x-y=4503.

$$79\left(-92y+5336\right)-y=4503$$

Multiply 79 times -92y+5336.

$$-7268y+421544-y=4503$$

Add -7268y to -y.

$$-7269y+421544=4503$$

Subtract 421544 from both sides of the equation.

$$-7269y=-417041$$

Divide both sides by -7269.

$$y=\frac{417041}{7269}$$

Substitute \frac{417041}{7269} for y in x=-92y+5336. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-92\times \frac{417041}{7269}+5336$$

Multiply -92 times \frac{417041}{7269}.

$$x=-\frac{38367772}{7269}+5336$$

Add 5336 to -\frac{38367772}{7269}.

$$x=\frac{419612}{7269}$$

The system is now solved.

$$x=\frac{419612}{7269},y=\frac{417041}{7269}$$

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 92.

$$x+92y=5336$$

Consider the second equation. Multiply both sides of the equation by 79.

$$79x-y=4503$$

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

$$x+92y=5336,79x-y=4503$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&92\\79&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5336\\4503\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&92\\79&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&92\\79&-1\end{matrix}\right))\left(\begin{matrix}1&92\\79&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&92\\79&-1\end{matrix}\right))\left(\begin{matrix}5336\\4503\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&92\\79&-1\end{matrix}\right))\left(\begin{matrix}5336\\4503\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&92\\79&-1\end{matrix}\right))\left(\begin{matrix}5336\\4503\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-92\times 79}&-\frac{92}{-1-92\times 79}\\-\frac{79}{-1-92\times 79}&\frac{1}{-1-92\times 79}\end{matrix}\right)\left(\begin{matrix}5336\\4503\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7269}&\frac{92}{7269}\\\frac{79}{7269}&-\frac{1}{7269}\end{matrix}\right)\left(\begin{matrix}5336\\4503\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7269}\times 5336+\frac{92}{7269}\times 4503\\\frac{79}{7269}\times 5336-\frac{1}{7269}\times 4503\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{419612}{7269}\\\frac{417041}{7269}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{419612}{7269},y=\frac{417041}{7269}$$

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 92.

$$x+92y=5336$$

Consider the second equation. Multiply both sides of the equation by 79.

$$79x-y=4503$$

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.

$$x+92y=5336,79x-y=4503$$

To make x and 79x equal, multiply all terms on each side of the first equation by 79 and all terms on each side of the second by 1.

$$79x+79\times 92y=79\times 5336,79x-y=4503$$

Simplify.

$$79x+7268y=421544,79x-y=4503$$

Subtract 79x-y=4503 from 79x+7268y=421544 by subtracting like terms on each side of the equal sign.

$$79x-79x+7268y+y=421544-4503$$

Add 79x to -79x. Terms 79x and -79x cancel out, leaving an equation with only one variable that can be solved.

$$7268y+y=421544-4503$$

Add 7268y to y.

$$7269y=421544-4503$$

Add 421544 to -4503.

$$7269y=417041$$

Divide both sides by 7269.

$$y=\frac{417041}{7269}$$

Substitute \frac{417041}{7269} for y in 79x-y=4503. Because the resulting equation contains only one variable, you can solve for x directly.

$$79x-\frac{417041}{7269}=4503$$

Add \frac{417041}{7269} to both sides of the equation.

$$79x=\frac{33149348}{7269}$$

Divide both sides by 79.

$$x=\frac{419612}{7269}$$

The system is now solved.

$$x=\frac{419612}{7269},y=\frac{417041}{7269}$$