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2x+5y=431,9x+7y=697
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=431
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
2x=-5y+431
Subtract 5y from both sides of the equation.
x=\frac{1}{2}\left(-5y+431\right)
Divide both sides by 2.
x=-\frac{5}{2}y+\frac{431}{2}
Multiply \frac{1}{2} times -5y+431.
9\left(-\frac{5}{2}y+\frac{431}{2}\right)+7y=697
Substitute \frac{-5y+431}{2} for x in the other equation, 9x+7y=697.
-\frac{45}{2}y+\frac{3879}{2}+7y=697
Multiply 9 times \frac{-5y+431}{2}.
-\frac{31}{2}y+\frac{3879}{2}=697
Add -\frac{45y}{2} to 7y.
-\frac{31}{2}y=-\frac{2485}{2}
Subtract \frac{3879}{2} from both sides of the equation.
y=\frac{2485}{31}
Divide both sides of the equation by -\frac{31}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.
x=-\frac{5}{2}\times \frac{2485}{31}+\frac{431}{2}
Substitute \frac{2485}{31} for y in x=-\frac{5}{2}y+\frac{431}{2}. Because the resulting equation contains only one variable, you can solve for x directly.
x=-\frac{12425}{62}+\frac{431}{2}
Multiply -\frac{5}{2} times \frac{2485}{31} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
x=\frac{468}{31}
Add \frac{431}{2} to -\frac{12425}{62} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=\frac{468}{31},y=\frac{2485}{31}
The system is now solved.
2x+5y=431,9x+7y=697
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}2&5\\9&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}431\\697\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}2&5\\9&7\end{matrix}\right))\left(\begin{matrix}2&5\\9&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\9&7\end{matrix}\right))\left(\begin{matrix}431\\697\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&5\\9&7\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\\9&7\end{matrix}\right))\left(\begin{matrix}431\\697\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\\9&7\end{matrix}\right))\left(\begin{matrix}431\\697\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{7}{2\times 7-5\times 9}&-\frac{5}{2\times 7-5\times 9}\\-\frac{9}{2\times 7-5\times 9}&\frac{2}{2\times 7-5\times 9}\end{matrix}\right)\left(\begin{matrix}431\\697\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{7}{31}&\frac{5}{31}\\\frac{9}{31}&-\frac{2}{31}\end{matrix}\right)\left(\begin{matrix}431\\697\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{31}\times 431+\frac{5}{31}\times 697\\\frac{9}{31}\times 431-\frac{2}{31}\times 697\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{468}{31}\\\frac{2485}{31}\end{matrix}\right)
Do the arithmetic.
x=\frac{468}{31},y=\frac{2485}{31}
Extract the matrix elements x and y.
2x+5y=431,9x+7y=697
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.
9\times 2x+9\times 5y=9\times 431,2\times 9x+2\times 7y=2\times 697
To make 2x and 9x equal, multiply all terms on each side of the first equation by 9 and all terms on each side of the second by 2.
18x+45y=3879,18x+14y=1394
Simplify.
18x-18x+45y-14y=3879-1394
Subtract 18x+14y=1394 from 18x+45y=3879 by subtracting like terms on each side of the equal sign.
45y-14y=3879-1394
Add 18x to -18x. Terms 18x and -18x cancel out, leaving an equation with only one variable that can be solved.
31y=3879-1394
Add 45y to -14y.
31y=2485
Add 3879 to -1394.
y=\frac{2485}{31}
Divide both sides by 31.
9x+7\times \frac{2485}{31}=697
Substitute \frac{2485}{31} for y in 9x+7y=697. Because the resulting equation contains only one variable, you can solve for x directly.
9x+\frac{17395}{31}=697
Multiply 7 times \frac{2485}{31}.
9x=\frac{4212}{31}
Subtract \frac{17395}{31} from both sides of the equation.
x=\frac{468}{31}
Divide both sides by 9.
x=\frac{468}{31},y=\frac{2485}{31}
The system is now solved.