Solve {l}{2x+5y=259}{199x-2y=1127}

$$\left. \begin{array} { l } { 2 x + 5 y = 259 } \\ { 199 x - 2 y = 1127 } \end{array} \right.$$

Solve for x, y

$$x = \frac{2051}{333} = 6\frac{53}{333} \approx 6.159159159$$

$$y = \frac{16429}{333} = 49\frac{112}{333} \approx 49.336336336$$

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.

$$2x+5y=259,199x-2y=1127$$

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

$$2x+5y=259$$

Subtract 5y from both sides of the equation.

$$2x=-5y+259$$

Divide both sides by 2.

$$x=\frac{1}{2}\left(-5y+259\right)$$

Multiply \frac{1}{2} times -5y+259.

$$x=-\frac{5}{2}y+\frac{259}{2}$$

Substitute \frac{-5y+259}{2} for x in the other equation, 199x-2y=1127.

$$199\left(-\frac{5}{2}y+\frac{259}{2}\right)-2y=1127$$

Multiply 199 times \frac{-5y+259}{2}.

$$-\frac{995}{2}y+\frac{51541}{2}-2y=1127$$

Add -\frac{995y}{2} to -2y.

$$-\frac{999}{2}y+\frac{51541}{2}=1127$$

Subtract \frac{51541}{2} from both sides of the equation.

$$-\frac{999}{2}y=-\frac{49287}{2}$$

Divide both sides of the equation by -\frac{999}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{16429}{333}$$

Substitute \frac{16429}{333} for y in x=-\frac{5}{2}y+\frac{259}{2}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{5}{2}\times \frac{16429}{333}+\frac{259}{2}$$

Multiply -\frac{5}{2} times \frac{16429}{333} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{82145}{666}+\frac{259}{2}$$

Add \frac{259}{2} to -\frac{82145}{666} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{2051}{333}$$

The system is now solved.

$$x=\frac{2051}{333},y=\frac{16429}{333}$$

Steps Using Matrices

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

$$2x+5y=259,199x-2y=1127$$

Write the equations in matrix form.

$$\left(\begin{matrix}2&5\\199&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}259\\1127\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&5\\199&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&5\\199&-2\end{matrix}\right))\left(\begin{matrix}2&5\\199&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\199&-2\end{matrix}\right))\left(\begin{matrix}259\\1127\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\199&-2\end{matrix}\right))\left(\begin{matrix}259\\1127\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\199&-2\end{matrix}\right))\left(\begin{matrix}259\\1127\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{2\left(-2\right)-5\times 199}&-\frac{5}{2\left(-2\right)-5\times 199}\\-\frac{199}{2\left(-2\right)-5\times 199}&\frac{2}{2\left(-2\right)-5\times 199}\end{matrix}\right)\left(\begin{matrix}259\\1127\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{999}&\frac{5}{999}\\\frac{199}{999}&-\frac{2}{999}\end{matrix}\right)\left(\begin{matrix}259\\1127\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{999}\times 259+\frac{5}{999}\times 1127\\\frac{199}{999}\times 259-\frac{2}{999}\times 1127\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2051}{333}\\\frac{16429}{333}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{2051}{333},y=\frac{16429}{333}$$

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.

$$2x+5y=259,199x-2y=1127$$

To make 2x and 199x equal, multiply all terms on each side of the first equation by 199 and all terms on each side of the second by 2.

$$199\times 2x+199\times 5y=199\times 259,2\times 199x+2\left(-2\right)y=2\times 1127$$

Simplify.

$$398x+995y=51541,398x-4y=2254$$

Subtract 398x-4y=2254 from 398x+995y=51541 by subtracting like terms on each side of the equal sign.

$$398x-398x+995y+4y=51541-2254$$

Add 398x to -398x. Terms 398x and -398x cancel out, leaving an equation with only one variable that can be solved.

$$995y+4y=51541-2254$$

Add 995y to 4y.

$$999y=51541-2254$$

Add 51541 to -2254.

$$999y=49287$$

Divide both sides by 999.

$$y=\frac{16429}{333}$$

Substitute \frac{16429}{333} for y in 199x-2y=1127. Because the resulting equation contains only one variable, you can solve for x directly.

$$199x-2\times \frac{16429}{333}=1127$$

Multiply -2 times \frac{16429}{333}.

$$199x-\frac{32858}{333}=1127$$

Add \frac{32858}{333} to both sides of the equation.

$$199x=\frac{408149}{333}$$

Divide both sides by 199.

$$x=\frac{2051}{333}$$

The system is now solved.

$$x=\frac{2051}{333},y=\frac{16429}{333}$$