Solve for x, y

Steps Using Substitution
Steps Using Matrices
Steps Using Elimination
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2x+5y=259,199x-2y=1127
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
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.
x=\frac{1}{2}\left(-5y+259\right)
Divide both sides by 2.
x=-\frac{5}{2}y+\frac{259}{2}
Multiply \frac{1}{2}=0.5 times -5y+259.
199\left(-\frac{5}{2}y+\frac{259}{2}\right)-2y=1127
Substitute \frac{-5y+259}{2} for x in the other equation, 199x-2y=1127.
-\frac{995}{2}y+\frac{51541}{2}-2y=1127
Multiply 199 times \frac{-5y+259}{2}.
-\frac{999}{2}y+\frac{51541}{2}=1127
-\frac{999}{2}y=-\frac{49287}{2}
Subtract \frac{51541}{2}=25770.5 from both sides of the equation.
y=\frac{16429}{333}
Divide both sides of the equation by -\frac{999}{2}=-499.5, which is the same as multiplying both sides by the reciprocal of the fraction.
x=-\frac{5}{2}\times \left(\frac{16429}{333}\right)+\frac{259}{2}
Substitute \frac{16429}{333}\approx 49.336336336 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{82145}{666}+\frac{259}{2}
Multiply -\frac{5}{2}=-2.5 times \frac{16429}{333}\approx 49.336336336 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
x=\frac{2051}{333}
Add \frac{259}{2}=129.5 to -\frac{82145}{666}\approx -123.340840841 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=\frac{2051}{333},y=\frac{16429}{333}
The system is now solved.
2x+5y=259,199x-2y=1127
Put the equations in standard form and then use matrices to solve the system of equations.
\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)
Write the equations in matrix form.
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)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&5\\199&-2\end{matrix}\right).
\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)
The product of a matrix and its inverse is the identity matrix.
\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)=\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)
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}{999}&\frac{5}{999}\\\frac{199}{999}&-\frac{2}{999}\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}\times 259+\frac{5}{999}\times 1127\\\frac{199}{999}\times 259-\frac{2}{999}\times 1127\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2051}{333}\\\frac{16429}{333}\end{matrix}\right)
Do the arithmetic.
x=\frac{2051}{333},y=\frac{16429}{333}
Extract the matrix elements x and y.
2x+5y=259,199x-2y=1127
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.
199\times 2x+199\times 5y=199\times 259,2\times 199x+2\left(-2\right)y=2\times 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.
398x+995y=51541,398x-4y=2254
Simplify.
398x-398x+995y+4y=51541-2254
Subtract 398x-4y=2254 from 398x+995y=51541 by subtracting like terms on each side of the equal sign.
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.
999y=51541-2254
999y=49287