Solve for x, y
x=\frac{1461920290375446110715}{17}\approx 8.59953112 \cdot 10^{19}
y=\frac{3654800725938615276626}{17}\approx 2.14988278 \cdot 10^{20}
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Simultaneous Equation5 problems similar to: \left. \begin{array} { l } { 7 x + 4 y = 13 ^ { 19 } } \\ { 5 x - 2 y = 19 } \end{array} \right.Similar Problems from Web Search
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7x+4y=1461920290375446110677,5x-2y=19
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.
7x+4y=1461920290375446110677
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
7x=-4y+1461920290375446110677
Subtract 4y from both sides of the equation.
x=\frac{1}{7}\left(-4y+1461920290375446110677\right)
Divide both sides by 7.
x=-\frac{4}{7}y+\frac{1461920290375446110677}{7}
Multiply \frac{1}{7} times -4y+1461920290375446110677.
5\left(-\frac{4}{7}y+\frac{1461920290375446110677}{7}\right)-2y=19
Substitute \frac{-4y+1461920290375446110677}{7} for x in the other equation, 5x-2y=19.
-\frac{20}{7}y+\frac{7309601451877230553385}{7}-2y=19
Multiply 5 times \frac{-4y+1461920290375446110677}{7}.
-\frac{34}{7}y+\frac{7309601451877230553385}{7}=19
Add -\frac{20y}{7} to -2y.
-\frac{34}{7}y=-\frac{7309601451877230553252}{7}
Subtract \frac{7309601451877230553385}{7} from both sides of the equation.
y=\frac{3654800725938615276626}{17}
Divide both sides of the equation by -\frac{34}{7}, which is the same as multiplying both sides by the reciprocal of the fraction.
x=-\frac{4}{7}\times \frac{3654800725938615276626}{17}+\frac{1461920290375446110677}{7}
Substitute \frac{3654800725938615276626}{17} for y in x=-\frac{4}{7}y+\frac{1461920290375446110677}{7}. Because the resulting equation contains only one variable, you can solve for x directly.
x=-\frac{14619202903754461106504}{119}+\frac{1461920290375446110677}{7}
Multiply -\frac{4}{7} times \frac{3654800725938615276626}{17} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
x=\frac{1461920290375446110715}{17}
Add \frac{1461920290375446110677}{7} to -\frac{14619202903754461106504}{119} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=\frac{1461920290375446110715}{17},y=\frac{3654800725938615276626}{17}
The system is now solved.
7x+4y=1461920290375446110677,5x-2y=19
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}7&4\\5&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1461920290375446110677\\19\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}7&4\\5&-2\end{matrix}\right))\left(\begin{matrix}7&4\\5&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&4\\5&-2\end{matrix}\right))\left(\begin{matrix}1461920290375446110677\\19\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}7&4\\5&-2\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}7&4\\5&-2\end{matrix}\right))\left(\begin{matrix}1461920290375446110677\\19\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}7&4\\5&-2\end{matrix}\right))\left(\begin{matrix}1461920290375446110677\\19\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{2}{7\left(-2\right)-4\times 5}&-\frac{4}{7\left(-2\right)-4\times 5}\\-\frac{5}{7\left(-2\right)-4\times 5}&\frac{7}{7\left(-2\right)-4\times 5}\end{matrix}\right)\left(\begin{matrix}1461920290375446110677\\19\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}{17}&\frac{2}{17}\\\frac{5}{34}&-\frac{7}{34}\end{matrix}\right)\left(\begin{matrix}1461920290375446110677\\19\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{17}\times 1461920290375446110677+\frac{2}{17}\times 19\\\frac{5}{34}\times 1461920290375446110677-\frac{7}{34}\times 19\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1461920290375446110715}{17}\\\frac{3654800725938615276626}{17}\end{matrix}\right)
Do the arithmetic.
x=\frac{1461920290375446110715}{17},y=\frac{3654800725938615276626}{17}
Extract the matrix elements x and y.
7x+4y=1461920290375446110677,5x-2y=19
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.
5\times 7x+5\times 4y=5\times 1461920290375446110677,7\times 5x+7\left(-2\right)y=7\times 19
To make 7x and 5x equal, multiply all terms on each side of the first equation by 5 and all terms on each side of the second by 7.
35x+20y=7309601451877230553385,35x-14y=133
Simplify.
35x-35x+20y+14y=7309601451877230553385-133
Subtract 35x-14y=133 from 35x+20y=7309601451877230553385 by subtracting like terms on each side of the equal sign.
20y+14y=7309601451877230553385-133
Add 35x to -35x. Terms 35x and -35x cancel out, leaving an equation with only one variable that can be solved.
34y=7309601451877230553385-133
Add 20y to 14y.
34y=7309601451877230553252
Add 7309601451877230553385 to -133.
y=\frac{3654800725938615276626}{17}
Divide both sides by 34.
5x-2\times \frac{3654800725938615276626}{17}=19
Substitute \frac{3654800725938615276626}{17} for y in 5x-2y=19. Because the resulting equation contains only one variable, you can solve for x directly.
5x-\frac{7309601451877230553252}{17}=19
Multiply -2 times \frac{3654800725938615276626}{17}.
5x=\frac{7309601451877230553575}{17}
Add \frac{7309601451877230553252}{17} to both sides of the equation.
x=\frac{1461920290375446110715}{17}
Divide both sides by 5.
x=\frac{1461920290375446110715}{17},y=\frac{3654800725938615276626}{17}
The system is now solved.