Solve {l}{2x+20y=1359.55}{12x+9y=641.7995}

$$\left. \begin{array} { l } { 2 x + 20 y = 1359.55 } \\ { 12 x + 9 y = 641.7995 } \end{array} \right.$$

Solve for x, y

$$x = \frac{15001}{5550} = 2\frac{3901}{5550} \approx 2.702882883$$

$$y = \frac{15031001}{222000} = 67\frac{157001}{222000} \approx 67.707211712$$

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.

$$2x+20y=1359.55,12x+9y=641.7995$$

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

$$2x+20y=1359.55$$

Subtract 20y from both sides of the equation.

$$2x=-20y+1359.55$$

Divide both sides by 2.

$$x=\frac{1}{2}\left(-20y+1359.55\right)$$

Multiply \frac{1}{2} times -20y+1359.55.

$$x=-10y+\frac{27191}{40}$$

Substitute -10y+\frac{27191}{40} for x in the other equation, 12x+9y=641.7995.

$$12\left(-10y+\frac{27191}{40}\right)+9y=641.7995$$

Multiply 12 times -10y+\frac{27191}{40}.

$$-120y+\frac{81573}{10}+9y=641.7995$$

Add -120y to 9y.

$$-111y+\frac{81573}{10}=641.7995$$

Subtract \frac{81573}{10} from both sides of the equation.

$$-111y=-\frac{15031001}{2000}$$

Divide both sides by -111.

$$y=\frac{15031001}{222000}$$

Substitute \frac{15031001}{222000} for y in x=-10y+\frac{27191}{40}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-10\times \frac{15031001}{222000}+\frac{27191}{40}$$

Multiply -10 times \frac{15031001}{222000}.

$$x=-\frac{15031001}{22200}+\frac{27191}{40}$$

Add \frac{27191}{40} to -\frac{15031001}{22200} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{15001}{5550}$$

The system is now solved.

$$x=\frac{15001}{5550},y=\frac{15031001}{222000}$$

Steps Using Matrices

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

$$2x+20y=1359.55,12x+9y=641.7995$$

Write the equations in matrix form.

$$\left(\begin{matrix}2&20\\12&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1359.55\\641.7995\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&20\\12&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&20\\12&9\end{matrix}\right))\left(\begin{matrix}2&20\\12&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&20\\12&9\end{matrix}\right))\left(\begin{matrix}1359.55\\641.7995\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&20\\12&9\end{matrix}\right))\left(\begin{matrix}1359.55\\641.7995\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&20\\12&9\end{matrix}\right))\left(\begin{matrix}1359.55\\641.7995\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{2\times 9-20\times 12}&-\frac{20}{2\times 9-20\times 12}\\-\frac{12}{2\times 9-20\times 12}&\frac{2}{2\times 9-20\times 12}\end{matrix}\right)\left(\begin{matrix}1359.55\\641.7995\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{74}&\frac{10}{111}\\\frac{2}{37}&-\frac{1}{111}\end{matrix}\right)\left(\begin{matrix}1359.55\\641.7995\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{74}\times 1359.55+\frac{10}{111}\times 641.7995\\\frac{2}{37}\times 1359.55-\frac{1}{111}\times 641.7995\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15001}{5550}\\\frac{15031001}{222000}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{15001}{5550},y=\frac{15031001}{222000}$$

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.

$$2x+20y=1359.55,12x+9y=641.7995$$

To make 2x and 12x equal, multiply all terms on each side of the first equation by 12 and all terms on each side of the second by 2.

$$12\times 2x+12\times 20y=12\times 1359.55,2\times 12x+2\times 9y=2\times 641.7995$$

Simplify.

$$24x+240y=16314.6,24x+18y=1283.599$$

Subtract 24x+18y=1283.599 from 24x+240y=16314.6 by subtracting like terms on each side of the equal sign.

$$24x-24x+240y-18y=16314.6-1283.599$$

Add 24x to -24x. Terms 24x and -24x cancel out, leaving an equation with only one variable that can be solved.

$$240y-18y=16314.6-1283.599$$

Add 240y to -18y.

$$222y=16314.6-1283.599$$

Add 16314.6 to -1283.599 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$222y=15031.001$$

Divide both sides by 222.

$$y=\frac{15031001}{222000}$$

Substitute \frac{15031001}{222000} for y in 12x+9y=641.7995. Because the resulting equation contains only one variable, you can solve for x directly.

$$12x+9\times \frac{15031001}{222000}=641.7995$$

Multiply 9 times \frac{15031001}{222000}.

$$12x+\frac{45093003}{74000}=641.7995$$

Subtract \frac{45093003}{74000} from both sides of the equation.

$$12x=\frac{30002}{925}$$

Divide both sides by 12.

$$x=\frac{15001}{5550}$$

The system is now solved.

$$x=\frac{15001}{5550},y=\frac{15031001}{222000}$$