Solve for x, y
x = -\frac{3861}{614} = -6\frac{177}{614} \approx -6.288273616
y=\frac{547}{614}\approx 0.890879479
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15x+107y=1,71x+179y=-287
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.
15x+107y=1
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
15x=-107y+1
Subtract 107y from both sides of the equation.
x=\frac{1}{15}\left(-107y+1\right)
Divide both sides by 15.
x=-\frac{107}{15}y+\frac{1}{15}
Multiply \frac{1}{15} times -107y+1.
71\left(-\frac{107}{15}y+\frac{1}{15}\right)+179y=-287
Substitute \frac{-107y+1}{15} for x in the other equation, 71x+179y=-287.
-\frac{7597}{15}y+\frac{71}{15}+179y=-287
Multiply 71 times \frac{-107y+1}{15}.
-\frac{4912}{15}y+\frac{71}{15}=-287
Add -\frac{7597y}{15} to 179y.
-\frac{4912}{15}y=-\frac{4376}{15}
Subtract \frac{71}{15} from both sides of the equation.
y=\frac{547}{614}
Divide both sides of the equation by -\frac{4912}{15}, which is the same as multiplying both sides by the reciprocal of the fraction.
x=-\frac{107}{15}\times \frac{547}{614}+\frac{1}{15}
Substitute \frac{547}{614} for y in x=-\frac{107}{15}y+\frac{1}{15}. Because the resulting equation contains only one variable, you can solve for x directly.
x=-\frac{58529}{9210}+\frac{1}{15}
Multiply -\frac{107}{15} times \frac{547}{614} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
x=-\frac{3861}{614}
Add \frac{1}{15} to -\frac{58529}{9210} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=-\frac{3861}{614},y=\frac{547}{614}
The system is now solved.
15x+107y=1,71x+179y=-287
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}15&107\\71&179\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\-287\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}15&107\\71&179\end{matrix}\right))\left(\begin{matrix}15&107\\71&179\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}15&107\\71&179\end{matrix}\right))\left(\begin{matrix}1\\-287\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}15&107\\71&179\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}15&107\\71&179\end{matrix}\right))\left(\begin{matrix}1\\-287\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}15&107\\71&179\end{matrix}\right))\left(\begin{matrix}1\\-287\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{179}{15\times 179-107\times 71}&-\frac{107}{15\times 179-107\times 71}\\-\frac{71}{15\times 179-107\times 71}&\frac{15}{15\times 179-107\times 71}\end{matrix}\right)\left(\begin{matrix}1\\-287\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{179}{4912}&\frac{107}{4912}\\\frac{71}{4912}&-\frac{15}{4912}\end{matrix}\right)\left(\begin{matrix}1\\-287\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{179}{4912}+\frac{107}{4912}\left(-287\right)\\\frac{71}{4912}-\frac{15}{4912}\left(-287\right)\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3861}{614}\\\frac{547}{614}\end{matrix}\right)
Do the arithmetic.
x=-\frac{3861}{614},y=\frac{547}{614}
Extract the matrix elements x and y.
15x+107y=1,71x+179y=-287
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.
71\times 15x+71\times 107y=71,15\times 71x+15\times 179y=15\left(-287\right)
To make 15x and 71x equal, multiply all terms on each side of the first equation by 71 and all terms on each side of the second by 15.
1065x+7597y=71,1065x+2685y=-4305
Simplify.
1065x-1065x+7597y-2685y=71+4305
Subtract 1065x+2685y=-4305 from 1065x+7597y=71 by subtracting like terms on each side of the equal sign.
7597y-2685y=71+4305
Add 1065x to -1065x. Terms 1065x and -1065x cancel out, leaving an equation with only one variable that can be solved.
4912y=71+4305
Add 7597y to -2685y.
4912y=4376
Add 71 to 4305.
y=\frac{547}{614}
Divide both sides by 4912.
71x+179\times \frac{547}{614}=-287
Substitute \frac{547}{614} for y in 71x+179y=-287. Because the resulting equation contains only one variable, you can solve for x directly.
71x+\frac{97913}{614}=-287
Multiply 179 times \frac{547}{614}.
71x=-\frac{274131}{614}
Subtract \frac{97913}{614} from both sides of the equation.
x=-\frac{3861}{614}
Divide both sides by 71.
x=-\frac{3861}{614},y=\frac{547}{614}
The system is now solved.
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Limits
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