Solve for x, y
x = \frac{2095227500}{127131} = 16480\frac{108620}{127131} \approx 16480.854394286
y = \frac{482642500}{127131} = 3796\frac{53224}{127131} \approx 3796.418654773
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21858x+16674y=423540000,9609x+382965y=1612260000
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.
21858x+16674y=423540000
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
21858x=-16674y+423540000
Subtract 16674y from both sides of the equation.
x=\frac{1}{21858}\left(-16674y+423540000\right)
Divide both sides by 21858.
x=-\frac{2779}{3643}y+\frac{70590000}{3643}
Multiply \frac{1}{21858} times -16674y+423540000.
9609\left(-\frac{2779}{3643}y+\frac{70590000}{3643}\right)+382965y=1612260000
Substitute \frac{-2779y+70590000}{3643} for x in the other equation, 9609x+382965y=1612260000.
-\frac{26703411}{3643}y+\frac{678299310000}{3643}+382965y=1612260000
Multiply 9609 times \frac{-2779y+70590000}{3643}.
\frac{1368438084}{3643}y+\frac{678299310000}{3643}=1612260000
Add -\frac{26703411y}{3643} to 382965y.
\frac{1368438084}{3643}y=\frac{5195163870000}{3643}
Subtract \frac{678299310000}{3643} from both sides of the equation.
y=\frac{482642500}{127131}
Divide both sides of the equation by \frac{1368438084}{3643}, which is the same as multiplying both sides by the reciprocal of the fraction.
x=-\frac{2779}{3643}\times \frac{482642500}{127131}+\frac{70590000}{3643}
Substitute \frac{482642500}{127131} for y in x=-\frac{2779}{3643}y+\frac{70590000}{3643}. Because the resulting equation contains only one variable, you can solve for x directly.
x=-\frac{1341263507500}{463138233}+\frac{70590000}{3643}
Multiply -\frac{2779}{3643} times \frac{482642500}{127131} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
x=\frac{2095227500}{127131}
Add \frac{70590000}{3643} to -\frac{1341263507500}{463138233} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=\frac{2095227500}{127131},y=\frac{482642500}{127131}
The system is now solved.
21858x+16674y=423540000,9609x+382965y=1612260000
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}21858&16674\\9609&382965\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}423540000\\1612260000\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}21858&16674\\9609&382965\end{matrix}\right))\left(\begin{matrix}21858&16674\\9609&382965\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}21858&16674\\9609&382965\end{matrix}\right))\left(\begin{matrix}423540000\\1612260000\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}21858&16674\\9609&382965\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}21858&16674\\9609&382965\end{matrix}\right))\left(\begin{matrix}423540000\\1612260000\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}21858&16674\\9609&382965\end{matrix}\right))\left(\begin{matrix}423540000\\1612260000\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{382965}{21858\times 382965-16674\times 9609}&-\frac{16674}{21858\times 382965-16674\times 9609}\\-\frac{9609}{21858\times 382965-16674\times 9609}&\frac{21858}{21858\times 382965-16674\times 9609}\end{matrix}\right)\left(\begin{matrix}423540000\\1612260000\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{127655}{2736876168}&-\frac{2779}{1368438084}\\-\frac{3203}{2736876168}&\frac{3643}{1368438084}\end{matrix}\right)\left(\begin{matrix}423540000\\1612260000\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{127655}{2736876168}\times 423540000-\frac{2779}{1368438084}\times 1612260000\\-\frac{3203}{2736876168}\times 423540000+\frac{3643}{1368438084}\times 1612260000\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2095227500}{127131}\\\frac{482642500}{127131}\end{matrix}\right)
Do the arithmetic.
x=\frac{2095227500}{127131},y=\frac{482642500}{127131}
Extract the matrix elements x and y.
21858x+16674y=423540000,9609x+382965y=1612260000
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.
9609\times 21858x+9609\times 16674y=9609\times 423540000,21858\times 9609x+21858\times 382965y=21858\times 1612260000
To make 21858x and 9609x equal, multiply all terms on each side of the first equation by 9609 and all terms on each side of the second by 21858.
210033522x+160220466y=4069795860000,210033522x+8370848970y=35240779080000
Simplify.
210033522x-210033522x+160220466y-8370848970y=4069795860000-35240779080000
Subtract 210033522x+8370848970y=35240779080000 from 210033522x+160220466y=4069795860000 by subtracting like terms on each side of the equal sign.
160220466y-8370848970y=4069795860000-35240779080000
Add 210033522x to -210033522x. Terms 210033522x and -210033522x cancel out, leaving an equation with only one variable that can be solved.
-8210628504y=4069795860000-35240779080000
Add 160220466y to -8370848970y.
-8210628504y=-31170983220000
Add 4069795860000 to -35240779080000.
y=\frac{482642500}{127131}
Divide both sides by -8210628504.
9609x+382965\times \frac{482642500}{127131}=1612260000
Substitute \frac{482642500}{127131} for y in 9609x+382965y=1612260000. Because the resulting equation contains only one variable, you can solve for x directly.
9609x+\frac{61611728337500}{42377}=1612260000
Multiply 382965 times \frac{482642500}{127131}.
9609x=\frac{6711013682500}{42377}
Subtract \frac{61611728337500}{42377} from both sides of the equation.
x=\frac{2095227500}{127131}
Divide both sides by 9609.
x=\frac{2095227500}{127131},y=\frac{482642500}{127131}
The system is now solved.
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Limits
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