\left\{ \begin{array} { l } { 1530 = 16 x + 2 y } \\ { 817 x + 110 y = 77715 } \end{array} \right.
Solve for x, y
x = \frac{715}{7} = 102\frac{1}{7} \approx 102.142857143
y = -\frac{365}{7} = -52\frac{1}{7} \approx -52.142857143
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16x+2y=1530
Consider the first equation. Swap sides so that all variable terms are on the left hand side.
16x+2y=1530,817x+110y=77715
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.
16x+2y=1530
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
16x=-2y+1530
Subtract 2y from both sides of the equation.
x=\frac{1}{16}\left(-2y+1530\right)
Divide both sides by 16.
x=-\frac{1}{8}y+\frac{765}{8}
Multiply \frac{1}{16} times -2y+1530.
817\left(-\frac{1}{8}y+\frac{765}{8}\right)+110y=77715
Substitute \frac{-y+765}{8} for x in the other equation, 817x+110y=77715.
-\frac{817}{8}y+\frac{625005}{8}+110y=77715
Multiply 817 times \frac{-y+765}{8}.
\frac{63}{8}y+\frac{625005}{8}=77715
Add -\frac{817y}{8} to 110y.
\frac{63}{8}y=-\frac{3285}{8}
Subtract \frac{625005}{8} from both sides of the equation.
y=-\frac{365}{7}
Divide both sides of the equation by \frac{63}{8}, which is the same as multiplying both sides by the reciprocal of the fraction.
x=-\frac{1}{8}\left(-\frac{365}{7}\right)+\frac{765}{8}
Substitute -\frac{365}{7} for y in x=-\frac{1}{8}y+\frac{765}{8}. Because the resulting equation contains only one variable, you can solve for x directly.
x=\frac{365}{56}+\frac{765}{8}
Multiply -\frac{1}{8} times -\frac{365}{7} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
x=\frac{715}{7}
Add \frac{765}{8} to \frac{365}{56} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=\frac{715}{7},y=-\frac{365}{7}
The system is now solved.
16x+2y=1530
Consider the first equation. Swap sides so that all variable terms are on the left hand side.
16x+2y=1530,817x+110y=77715
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}16&2\\817&110\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1530\\77715\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}16&2\\817&110\end{matrix}\right))\left(\begin{matrix}16&2\\817&110\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}16&2\\817&110\end{matrix}\right))\left(\begin{matrix}1530\\77715\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}16&2\\817&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}16&2\\817&110\end{matrix}\right))\left(\begin{matrix}1530\\77715\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}16&2\\817&110\end{matrix}\right))\left(\begin{matrix}1530\\77715\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}{16\times 110-2\times 817}&-\frac{2}{16\times 110-2\times 817}\\-\frac{817}{16\times 110-2\times 817}&\frac{16}{16\times 110-2\times 817}\end{matrix}\right)\left(\begin{matrix}1530\\77715\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{55}{63}&-\frac{1}{63}\\-\frac{817}{126}&\frac{8}{63}\end{matrix}\right)\left(\begin{matrix}1530\\77715\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{55}{63}\times 1530-\frac{1}{63}\times 77715\\-\frac{817}{126}\times 1530+\frac{8}{63}\times 77715\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{715}{7}\\-\frac{365}{7}\end{matrix}\right)
Do the arithmetic.
x=\frac{715}{7},y=-\frac{365}{7}
Extract the matrix elements x and y.
16x+2y=1530
Consider the first equation. Swap sides so that all variable terms are on the left hand side.
16x+2y=1530,817x+110y=77715
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.
817\times 16x+817\times 2y=817\times 1530,16\times 817x+16\times 110y=16\times 77715
To make 16x and 817x equal, multiply all terms on each side of the first equation by 817 and all terms on each side of the second by 16.
13072x+1634y=1250010,13072x+1760y=1243440
Simplify.
13072x-13072x+1634y-1760y=1250010-1243440
Subtract 13072x+1760y=1243440 from 13072x+1634y=1250010 by subtracting like terms on each side of the equal sign.
1634y-1760y=1250010-1243440
Add 13072x to -13072x. Terms 13072x and -13072x cancel out, leaving an equation with only one variable that can be solved.
-126y=1250010-1243440
Add 1634y to -1760y.
-126y=6570
Add 1250010 to -1243440.
y=-\frac{365}{7}
Divide both sides by -126.
817x+110\left(-\frac{365}{7}\right)=77715
Substitute -\frac{365}{7} for y in 817x+110y=77715. Because the resulting equation contains only one variable, you can solve for x directly.
817x-\frac{40150}{7}=77715
Multiply 110 times -\frac{365}{7}.
817x=\frac{584155}{7}
Add \frac{40150}{7} to both sides of the equation.
x=\frac{715}{7}
Divide both sides by 817.
x=\frac{715}{7},y=-\frac{365}{7}
The system is now solved.
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