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10a_{0}+66a_{1}=1194,66a_{0}+506a_{1}=7280
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.
10a_{0}+66a_{1}=1194
Choose one of the equations and solve it for a_{0} by isolating a_{0} on the left hand side of the equal sign.
10a_{0}=-66a_{1}+1194
Subtract 66a_{1} from both sides of the equation.
a_{0}=\frac{1}{10}\left(-66a_{1}+1194\right)
Divide both sides by 10.
a_{0}=-\frac{33}{5}a_{1}+\frac{597}{5}
Multiply \frac{1}{10} times -66a_{1}+1194.
66\left(-\frac{33}{5}a_{1}+\frac{597}{5}\right)+506a_{1}=7280
Substitute \frac{-33a_{1}+597}{5} for a_{0} in the other equation, 66a_{0}+506a_{1}=7280.
-\frac{2178}{5}a_{1}+\frac{39402}{5}+506a_{1}=7280
Multiply 66 times \frac{-33a_{1}+597}{5}.
\frac{352}{5}a_{1}+\frac{39402}{5}=7280
Add -\frac{2178a_{1}}{5} to 506a_{1}.
\frac{352}{5}a_{1}=-\frac{3002}{5}
Subtract \frac{39402}{5} from both sides of the equation.
a_{1}=-\frac{1501}{176}
Divide both sides of the equation by \frac{352}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.
a_{0}=-\frac{33}{5}\left(-\frac{1501}{176}\right)+\frac{597}{5}
Substitute -\frac{1501}{176} for a_{1} in a_{0}=-\frac{33}{5}a_{1}+\frac{597}{5}. Because the resulting equation contains only one variable, you can solve for a_{0} directly.
a_{0}=\frac{4503}{80}+\frac{597}{5}
Multiply -\frac{33}{5} times -\frac{1501}{176} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
a_{0}=\frac{2811}{16}
Add \frac{597}{5} to \frac{4503}{80} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
a_{0}=\frac{2811}{16},a_{1}=-\frac{1501}{176}
The system is now solved.
10a_{0}+66a_{1}=1194,66a_{0}+506a_{1}=7280
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}10&66\\66&506\end{matrix}\right)\left(\begin{matrix}a_{0}\\a_{1}\end{matrix}\right)=\left(\begin{matrix}1194\\7280\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}10&66\\66&506\end{matrix}\right))\left(\begin{matrix}10&66\\66&506\end{matrix}\right)\left(\begin{matrix}a_{0}\\a_{1}\end{matrix}\right)=inverse(\left(\begin{matrix}10&66\\66&506\end{matrix}\right))\left(\begin{matrix}1194\\7280\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}10&66\\66&506\end{matrix}\right).
\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a_{0}\\a_{1}\end{matrix}\right)=inverse(\left(\begin{matrix}10&66\\66&506\end{matrix}\right))\left(\begin{matrix}1194\\7280\end{matrix}\right)
The product of a matrix and its inverse is the identity matrix.
\left(\begin{matrix}a_{0}\\a_{1}\end{matrix}\right)=inverse(\left(\begin{matrix}10&66\\66&506\end{matrix}\right))\left(\begin{matrix}1194\\7280\end{matrix}\right)
Multiply the matrices on the left hand side of the equal sign.
\left(\begin{matrix}a_{0}\\a_{1}\end{matrix}\right)=\left(\begin{matrix}\frac{506}{10\times 506-66\times 66}&-\frac{66}{10\times 506-66\times 66}\\-\frac{66}{10\times 506-66\times 66}&\frac{10}{10\times 506-66\times 66}\end{matrix}\right)\left(\begin{matrix}1194\\7280\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}a_{0}\\a_{1}\end{matrix}\right)=\left(\begin{matrix}\frac{23}{32}&-\frac{3}{32}\\-\frac{3}{32}&\frac{5}{352}\end{matrix}\right)\left(\begin{matrix}1194\\7280\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}a_{0}\\a_{1}\end{matrix}\right)=\left(\begin{matrix}\frac{23}{32}\times 1194-\frac{3}{32}\times 7280\\-\frac{3}{32}\times 1194+\frac{5}{352}\times 7280\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}a_{0}\\a_{1}\end{matrix}\right)=\left(\begin{matrix}\frac{2811}{16}\\-\frac{1501}{176}\end{matrix}\right)
Do the arithmetic.
a_{0}=\frac{2811}{16},a_{1}=-\frac{1501}{176}
Extract the matrix elements a_{0} and a_{1}.
10a_{0}+66a_{1}=1194,66a_{0}+506a_{1}=7280
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.
66\times 10a_{0}+66\times 66a_{1}=66\times 1194,10\times 66a_{0}+10\times 506a_{1}=10\times 7280
To make 10a_{0} and 66a_{0} equal, multiply all terms on each side of the first equation by 66 and all terms on each side of the second by 10.
660a_{0}+4356a_{1}=78804,660a_{0}+5060a_{1}=72800
Simplify.
660a_{0}-660a_{0}+4356a_{1}-5060a_{1}=78804-72800
Subtract 660a_{0}+5060a_{1}=72800 from 660a_{0}+4356a_{1}=78804 by subtracting like terms on each side of the equal sign.
4356a_{1}-5060a_{1}=78804-72800
Add 660a_{0} to -660a_{0}. Terms 660a_{0} and -660a_{0} cancel out, leaving an equation with only one variable that can be solved.
-704a_{1}=78804-72800
Add 4356a_{1} to -5060a_{1}.
-704a_{1}=6004
Add 78804 to -72800.
a_{1}=-\frac{1501}{176}
Divide both sides by -704.
66a_{0}+506\left(-\frac{1501}{176}\right)=7280
Substitute -\frac{1501}{176} for a_{1} in 66a_{0}+506a_{1}=7280. Because the resulting equation contains only one variable, you can solve for a_{0} directly.
66a_{0}-\frac{34523}{8}=7280
Multiply 506 times -\frac{1501}{176}.
66a_{0}=\frac{92763}{8}
Add \frac{34523}{8} to both sides of the equation.
a_{0}=\frac{2811}{16}
Divide both sides by 66.
a_{0}=\frac{2811}{16},a_{1}=-\frac{1501}{176}
The system is now solved.