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10x+20y=\frac{14000}{1.4}
Consider the first equation. Divide both sides by 1.4.
10x+20y=\frac{140000}{14}
Expand \frac{14000}{1.4} by multiplying both numerator and the denominator by 10.
10x+20y=10000
Divide 140000 by 14 to get 10000.
110x+110y=\frac{89100}{1.1}
Consider the second equation. Divide both sides by 1.1.
110x+110y=\frac{891000}{11}
Expand \frac{89100}{1.1} by multiplying both numerator and the denominator by 10.
110x+110y=81000
Divide 891000 by 11 to get 81000.
10x+20y=10000,110x+110y=81000
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.
10x+20y=10000
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
10x=-20y+10000
Subtract 20y from both sides of the equation.
x=\frac{1}{10}\left(-20y+10000\right)
Divide both sides by 10.
x=-2y+1000
Multiply \frac{1}{10} times -20y+10000.
110\left(-2y+1000\right)+110y=81000
Substitute -2y+1000 for x in the other equation, 110x+110y=81000.
-220y+110000+110y=81000
Multiply 110 times -2y+1000.
-110y+110000=81000
Add -220y to 110y.
-110y=-29000
Subtract 110000 from both sides of the equation.
y=\frac{2900}{11}
Divide both sides by -110.
x=-2\times \frac{2900}{11}+1000
Substitute \frac{2900}{11} for y in x=-2y+1000. Because the resulting equation contains only one variable, you can solve for x directly.
x=-\frac{5800}{11}+1000
Multiply -2 times \frac{2900}{11}.
x=\frac{5200}{11}
Add 1000 to -\frac{5800}{11}.
x=\frac{5200}{11},y=\frac{2900}{11}
The system is now solved.
10x+20y=\frac{14000}{1.4}
Consider the first equation. Divide both sides by 1.4.
10x+20y=\frac{140000}{14}
Expand \frac{14000}{1.4} by multiplying both numerator and the denominator by 10.
10x+20y=10000
Divide 140000 by 14 to get 10000.
110x+110y=\frac{89100}{1.1}
Consider the second equation. Divide both sides by 1.1.
110x+110y=\frac{891000}{11}
Expand \frac{89100}{1.1} by multiplying both numerator and the denominator by 10.
110x+110y=81000
Divide 891000 by 11 to get 81000.
10x+20y=10000,110x+110y=81000
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}10&20\\110&110\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10000\\81000\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}10&20\\110&110\end{matrix}\right))\left(\begin{matrix}10&20\\110&110\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&20\\110&110\end{matrix}\right))\left(\begin{matrix}10000\\81000\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}10&20\\110&110\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}10&20\\110&110\end{matrix}\right))\left(\begin{matrix}10000\\81000\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}10&20\\110&110\end{matrix}\right))\left(\begin{matrix}10000\\81000\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{110}{10\times 110-20\times 110}&-\frac{20}{10\times 110-20\times 110}\\-\frac{110}{10\times 110-20\times 110}&\frac{10}{10\times 110-20\times 110}\end{matrix}\right)\left(\begin{matrix}10000\\81000\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{1}{10}&\frac{1}{55}\\\frac{1}{10}&-\frac{1}{110}\end{matrix}\right)\left(\begin{matrix}10000\\81000\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{10}\times 10000+\frac{1}{55}\times 81000\\\frac{1}{10}\times 10000-\frac{1}{110}\times 81000\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5200}{11}\\\frac{2900}{11}\end{matrix}\right)
Do the arithmetic.
x=\frac{5200}{11},y=\frac{2900}{11}
Extract the matrix elements x and y.
10x+20y=\frac{14000}{1.4}
Consider the first equation. Divide both sides by 1.4.
10x+20y=\frac{140000}{14}
Expand \frac{14000}{1.4} by multiplying both numerator and the denominator by 10.
10x+20y=10000
Divide 140000 by 14 to get 10000.
110x+110y=\frac{89100}{1.1}
Consider the second equation. Divide both sides by 1.1.
110x+110y=\frac{891000}{11}
Expand \frac{89100}{1.1} by multiplying both numerator and the denominator by 10.
110x+110y=81000
Divide 891000 by 11 to get 81000.
10x+20y=10000,110x+110y=81000
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.
110\times 10x+110\times 20y=110\times 10000,10\times 110x+10\times 110y=10\times 81000
To make 10x and 110x equal, multiply all terms on each side of the first equation by 110 and all terms on each side of the second by 10.
1100x+2200y=1100000,1100x+1100y=810000
Simplify.
1100x-1100x+2200y-1100y=1100000-810000
Subtract 1100x+1100y=810000 from 1100x+2200y=1100000 by subtracting like terms on each side of the equal sign.
2200y-1100y=1100000-810000
Add 1100x to -1100x. Terms 1100x and -1100x cancel out, leaving an equation with only one variable that can be solved.
1100y=1100000-810000
Add 2200y to -1100y.
1100y=290000
Add 1100000 to -810000.
y=\frac{2900}{11}
Divide both sides by 1100.
110x+110\times \frac{2900}{11}=81000
Substitute \frac{2900}{11} for y in 110x+110y=81000. Because the resulting equation contains only one variable, you can solve for x directly.
110x+29000=81000
Multiply 110 times \frac{2900}{11}.
110x=52000
Subtract 29000 from both sides of the equation.
x=\frac{5200}{11}
Divide both sides by 110.
x=\frac{5200}{11},y=\frac{2900}{11}
The system is now solved.