Solve for x, y
x = \frac{229736}{3195} = 71\frac{2891}{3195} \approx 71.90485133
y = -\frac{10336}{3195} = -3\frac{751}{3195} \approx -3.235054773
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x-94y=376
Consider the first equation. Multiply both sides of the equation by 94.
34x-y=2448
Consider the second equation. Multiply both sides of the equation by 34.
x-94y=376,34x-y=2448
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.
x-94y=376
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
x=94y+376
Add 94y to both sides of the equation.
34\left(94y+376\right)-y=2448
Substitute 376+94y for x in the other equation, 34x-y=2448.
3196y+12784-y=2448
Multiply 34 times 376+94y.
3195y+12784=2448
Add 3196y to -y.
3195y=-10336
Subtract 12784 from both sides of the equation.
y=-\frac{10336}{3195}
Divide both sides by 3195.
x=94\left(-\frac{10336}{3195}\right)+376
Substitute -\frac{10336}{3195} for y in x=94y+376. Because the resulting equation contains only one variable, you can solve for x directly.
x=-\frac{971584}{3195}+376
Multiply 94 times -\frac{10336}{3195}.
x=\frac{229736}{3195}
Add 376 to -\frac{971584}{3195}.
x=\frac{229736}{3195},y=-\frac{10336}{3195}
The system is now solved.
x-94y=376
Consider the first equation. Multiply both sides of the equation by 94.
34x-y=2448
Consider the second equation. Multiply both sides of the equation by 34.
x-94y=376,34x-y=2448
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}1&-94\\34&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}376\\2448\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}1&-94\\34&-1\end{matrix}\right))\left(\begin{matrix}1&-94\\34&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-94\\34&-1\end{matrix}\right))\left(\begin{matrix}376\\2448\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-94\\34&-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&-94\\34&-1\end{matrix}\right))\left(\begin{matrix}376\\2448\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&-94\\34&-1\end{matrix}\right))\left(\begin{matrix}376\\2448\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-\left(-94\times 34\right)}&-\frac{-94}{-1-\left(-94\times 34\right)}\\-\frac{34}{-1-\left(-94\times 34\right)}&\frac{1}{-1-\left(-94\times 34\right)}\end{matrix}\right)\left(\begin{matrix}376\\2448\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}{3195}&\frac{94}{3195}\\-\frac{34}{3195}&\frac{1}{3195}\end{matrix}\right)\left(\begin{matrix}376\\2448\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3195}\times 376+\frac{94}{3195}\times 2448\\-\frac{34}{3195}\times 376+\frac{1}{3195}\times 2448\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{229736}{3195}\\-\frac{10336}{3195}\end{matrix}\right)
Do the arithmetic.
x=\frac{229736}{3195},y=-\frac{10336}{3195}
Extract the matrix elements x and y.
x-94y=376
Consider the first equation. Multiply both sides of the equation by 94.
34x-y=2448
Consider the second equation. Multiply both sides of the equation by 34.
x-94y=376,34x-y=2448
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.
34x+34\left(-94\right)y=34\times 376,34x-y=2448
To make x and 34x equal, multiply all terms on each side of the first equation by 34 and all terms on each side of the second by 1.
34x-3196y=12784,34x-y=2448
Simplify.
34x-34x-3196y+y=12784-2448
Subtract 34x-y=2448 from 34x-3196y=12784 by subtracting like terms on each side of the equal sign.
-3196y+y=12784-2448
Add 34x to -34x. Terms 34x and -34x cancel out, leaving an equation with only one variable that can be solved.
-3195y=12784-2448
Add -3196y to y.
-3195y=10336
Add 12784 to -2448.
y=-\frac{10336}{3195}
Divide both sides by -3195.
34x-\left(-\frac{10336}{3195}\right)=2448
Substitute -\frac{10336}{3195} for y in 34x-y=2448. Because the resulting equation contains only one variable, you can solve for x directly.
34x=\frac{7811024}{3195}
Subtract \frac{10336}{3195} from both sides of the equation.
x=\frac{229736}{3195}
Divide both sides by 34.
x=\frac{229736}{3195},y=-\frac{10336}{3195}
The system is now solved.
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Limits
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