Solve {l}{x+y=53294}{18x+30y=1103748}

$$\left\{ \begin{array} { l } { x + y = 53294 } \\ { 18 x + 30 y = 1103748 } \end{array} \right.$$

Solve for x, y

$$x=41256$$

$$y=12038$$

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.

$$x+y=53294,18x+30y=1103748$$

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

$$x+y=53294$$

Subtract y from both sides of the equation.

$$x=-y+53294$$

Substitute -y+53294 for x in the other equation, 18x+30y=1103748.

$$18\left(-y+53294\right)+30y=1103748$$

Multiply 18 times -y+53294.

$$-18y+959292+30y=1103748$$

Add -18y to 30y.

$$12y+959292=1103748$$

Subtract 959292 from both sides of the equation.

$$12y=144456$$

Divide both sides by 12.

$$y=12038$$

Substitute 12038 for y in x=-y+53294. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-12038+53294$$

Add 53294 to -12038.

$$x=41256$$

The system is now solved.

$$x=41256,y=12038$$

Steps Using Matrices

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

$$x+y=53294,18x+30y=1103748$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&1\\18&30\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}53294\\1103748\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\18&30\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\18&30\end{matrix}\right))\left(\begin{matrix}1&1\\18&30\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\18&30\end{matrix}\right))\left(\begin{matrix}53294\\1103748\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\18&30\end{matrix}\right))\left(\begin{matrix}53294\\1103748\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\18&30\end{matrix}\right))\left(\begin{matrix}53294\\1103748\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{30}{30-18}&-\frac{1}{30-18}\\-\frac{18}{30-18}&\frac{1}{30-18}\end{matrix}\right)\left(\begin{matrix}53294\\1103748\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{2}&-\frac{1}{12}\\-\frac{3}{2}&\frac{1}{12}\end{matrix}\right)\left(\begin{matrix}53294\\1103748\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{2}\times 53294-\frac{1}{12}\times 1103748\\-\frac{3}{2}\times 53294+\frac{1}{12}\times 1103748\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}41256\\12038\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=41256,y=12038$$

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.

$$x+y=53294,18x+30y=1103748$$

To make x and 18x equal, multiply all terms on each side of the first equation by 18 and all terms on each side of the second by 1.

$$18x+18y=18\times 53294,18x+30y=1103748$$

Simplify.

$$18x+18y=959292,18x+30y=1103748$$

Subtract 18x+30y=1103748 from 18x+18y=959292 by subtracting like terms on each side of the equal sign.

$$18x-18x+18y-30y=959292-1103748$$

Add 18x to -18x. Terms 18x and -18x cancel out, leaving an equation with only one variable that can be solved.

$$18y-30y=959292-1103748$$

Add 18y to -30y.

$$-12y=959292-1103748$$

Add 959292 to -1103748.

$$-12y=-144456$$

Divide both sides by -12.

$$y=12038$$

Substitute 12038 for y in 18x+30y=1103748. Because the resulting equation contains only one variable, you can solve for x directly.

$$18x+30\times 12038=1103748$$

Multiply 30 times 12038.

$$18x+361140=1103748$$

Subtract 361140 from both sides of the equation.

$$18x=742608$$

Divide both sides by 18.

$$x=41256$$

The system is now solved.

$$x=41256,y=12038$$