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1.6x+0.5y=12250,x+y=1220
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.
1.6x+0.5y=12250
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
1.6x=-0.5y+12250
Subtract \frac{y}{2} from both sides of the equation.
x=0.625\left(-0.5y+12250\right)
Divide both sides of the equation by 1.6, which is the same as multiplying both sides by the reciprocal of the fraction.
x=-0.3125y+7656.25
Multiply 0.625 times -\frac{y}{2}+12250.
-0.3125y+7656.25+y=1220
Substitute -\frac{5y}{16}+7656.25 for x in the other equation, x+y=1220.
0.6875y+7656.25=1220
Add -\frac{5y}{16} to y.
0.6875y=-6436.25
Subtract 7656.25 from both sides of the equation.
y=-\frac{102980}{11}
Divide both sides of the equation by 0.6875, which is the same as multiplying both sides by the reciprocal of the fraction.
x=-0.3125\left(-\frac{102980}{11}\right)+7656.25
Substitute -\frac{102980}{11} for y in x=-0.3125y+7656.25. Because the resulting equation contains only one variable, you can solve for x directly.
x=\frac{128725}{44}+7656.25
Multiply -0.3125 times -\frac{102980}{11} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
x=\frac{116400}{11}
Add 7656.25 to \frac{128725}{44} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=\frac{116400}{11},y=-\frac{102980}{11}
The system is now solved.
1.6x+0.5y=12250,x+y=1220
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}1.6&0.5\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12250\\1220\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}1.6&0.5\\1&1\end{matrix}\right))\left(\begin{matrix}1.6&0.5\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1.6&0.5\\1&1\end{matrix}\right))\left(\begin{matrix}12250\\1220\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}1.6&0.5\\1&1\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}1.6&0.5\\1&1\end{matrix}\right))\left(\begin{matrix}12250\\1220\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}1.6&0.5\\1&1\end{matrix}\right))\left(\begin{matrix}12250\\1220\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{1}{1.6-0.5}&-\frac{0.5}{1.6-0.5}\\-\frac{1}{1.6-0.5}&\frac{1.6}{1.6-0.5}\end{matrix}\right)\left(\begin{matrix}12250\\1220\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{10}{11}&-\frac{5}{11}\\-\frac{10}{11}&\frac{16}{11}\end{matrix}\right)\left(\begin{matrix}12250\\1220\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{11}\times 12250-\frac{5}{11}\times 1220\\-\frac{10}{11}\times 12250+\frac{16}{11}\times 1220\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{116400}{11}\\-\frac{102980}{11}\end{matrix}\right)
Do the arithmetic.
x=\frac{116400}{11},y=-\frac{102980}{11}
Extract the matrix elements x and y.
1.6x+0.5y=12250,x+y=1220
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.
1.6x+0.5y=12250,1.6x+1.6y=1.6\times 1220
To make \frac{8x}{5} and x equal, multiply all terms on each side of the first equation by 1 and all terms on each side of the second by 1.6.
1.6x+0.5y=12250,1.6x+1.6y=1952
Simplify.
1.6x-1.6x+0.5y-1.6y=12250-1952
Subtract 1.6x+1.6y=1952 from 1.6x+0.5y=12250 by subtracting like terms on each side of the equal sign.
0.5y-1.6y=12250-1952
Add \frac{8x}{5} to -\frac{8x}{5}. Terms \frac{8x}{5} and -\frac{8x}{5} cancel out, leaving an equation with only one variable that can be solved.
-1.1y=12250-1952
Add \frac{y}{2} to -\frac{8y}{5}.
-1.1y=10298
Add 12250 to -1952.
y=-\frac{102980}{11}
Divide both sides of the equation by -1.1, which is the same as multiplying both sides by the reciprocal of the fraction.
x-\frac{102980}{11}=1220
Substitute -\frac{102980}{11} for y in x+y=1220. Because the resulting equation contains only one variable, you can solve for x directly.
x=\frac{116400}{11}
Add \frac{102980}{11} to both sides of the equation.
x=\frac{116400}{11},y=-\frac{102980}{11}
The system is now solved.