\left\{ \begin{array} { l } { ( 3 x - 2 y ) + \frac { 3 x + 5 y } { 7 } = - 50 } \\ { ( 3 x - 2 y ) - \frac { 3 x + 5 y } { 7 } = 5 } \end{array} \right.
Solve for x, y
x = -\frac{995}{42} = -23\frac{29}{42} \approx -23.69047619
y = -\frac{170}{7} = -24\frac{2}{7} \approx -24.285714286
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21x-14y+3x+5y=-350
Consider the first equation. Multiply both sides of the equation by 7.
24x-14y+5y=-350
Combine 21x and 3x to get 24x.
24x-9y=-350
Combine -14y and 5y to get -9y.
21x-14y-\left(3x+5y\right)=35
Consider the second equation. Multiply both sides of the equation by 7.
21x-14y-3x-5y=35
To find the opposite of 3x+5y, find the opposite of each term.
18x-14y-5y=35
Combine 21x and -3x to get 18x.
18x-19y=35
Combine -14y and -5y to get -19y.
24x-9y=-350,18x-19y=35
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.
24x-9y=-350
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
24x=9y-350
Add 9y to both sides of the equation.
x=\frac{1}{24}\left(9y-350\right)
Divide both sides by 24.
x=\frac{3}{8}y-\frac{175}{12}
Multiply \frac{1}{24} times 9y-350.
18\left(\frac{3}{8}y-\frac{175}{12}\right)-19y=35
Substitute \frac{3y}{8}-\frac{175}{12} for x in the other equation, 18x-19y=35.
\frac{27}{4}y-\frac{525}{2}-19y=35
Multiply 18 times \frac{3y}{8}-\frac{175}{12}.
-\frac{49}{4}y-\frac{525}{2}=35
Add \frac{27y}{4} to -19y.
-\frac{49}{4}y=\frac{595}{2}
Add \frac{525}{2} to both sides of the equation.
y=-\frac{170}{7}
Divide both sides of the equation by -\frac{49}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.
x=\frac{3}{8}\left(-\frac{170}{7}\right)-\frac{175}{12}
Substitute -\frac{170}{7} for y in x=\frac{3}{8}y-\frac{175}{12}. Because the resulting equation contains only one variable, you can solve for x directly.
x=-\frac{255}{28}-\frac{175}{12}
Multiply \frac{3}{8} times -\frac{170}{7} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
x=-\frac{995}{42}
Add -\frac{175}{12} to -\frac{255}{28} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=-\frac{995}{42},y=-\frac{170}{7}
The system is now solved.
21x-14y+3x+5y=-350
Consider the first equation. Multiply both sides of the equation by 7.
24x-14y+5y=-350
Combine 21x and 3x to get 24x.
24x-9y=-350
Combine -14y and 5y to get -9y.
21x-14y-\left(3x+5y\right)=35
Consider the second equation. Multiply both sides of the equation by 7.
21x-14y-3x-5y=35
To find the opposite of 3x+5y, find the opposite of each term.
18x-14y-5y=35
Combine 21x and -3x to get 18x.
18x-19y=35
Combine -14y and -5y to get -19y.
24x-9y=-350,18x-19y=35
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}24&-9\\18&-19\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-350\\35\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}24&-9\\18&-19\end{matrix}\right))\left(\begin{matrix}24&-9\\18&-19\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}24&-9\\18&-19\end{matrix}\right))\left(\begin{matrix}-350\\35\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}24&-9\\18&-19\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}24&-9\\18&-19\end{matrix}\right))\left(\begin{matrix}-350\\35\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}24&-9\\18&-19\end{matrix}\right))\left(\begin{matrix}-350\\35\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{19}{24\left(-19\right)-\left(-9\times 18\right)}&-\frac{-9}{24\left(-19\right)-\left(-9\times 18\right)}\\-\frac{18}{24\left(-19\right)-\left(-9\times 18\right)}&\frac{24}{24\left(-19\right)-\left(-9\times 18\right)}\end{matrix}\right)\left(\begin{matrix}-350\\35\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{19}{294}&-\frac{3}{98}\\\frac{3}{49}&-\frac{4}{49}\end{matrix}\right)\left(\begin{matrix}-350\\35\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{19}{294}\left(-350\right)-\frac{3}{98}\times 35\\\frac{3}{49}\left(-350\right)-\frac{4}{49}\times 35\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{995}{42}\\-\frac{170}{7}\end{matrix}\right)
Do the arithmetic.
x=-\frac{995}{42},y=-\frac{170}{7}
Extract the matrix elements x and y.
21x-14y+3x+5y=-350
Consider the first equation. Multiply both sides of the equation by 7.
24x-14y+5y=-350
Combine 21x and 3x to get 24x.
24x-9y=-350
Combine -14y and 5y to get -9y.
21x-14y-\left(3x+5y\right)=35
Consider the second equation. Multiply both sides of the equation by 7.
21x-14y-3x-5y=35
To find the opposite of 3x+5y, find the opposite of each term.
18x-14y-5y=35
Combine 21x and -3x to get 18x.
18x-19y=35
Combine -14y and -5y to get -19y.
24x-9y=-350,18x-19y=35
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.
18\times 24x+18\left(-9\right)y=18\left(-350\right),24\times 18x+24\left(-19\right)y=24\times 35
To make 24x and 18x equal, multiply all terms on each side of the first equation by 18 and all terms on each side of the second by 24.
432x-162y=-6300,432x-456y=840
Simplify.
432x-432x-162y+456y=-6300-840
Subtract 432x-456y=840 from 432x-162y=-6300 by subtracting like terms on each side of the equal sign.
-162y+456y=-6300-840
Add 432x to -432x. Terms 432x and -432x cancel out, leaving an equation with only one variable that can be solved.
294y=-6300-840
Add -162y to 456y.
294y=-7140
Add -6300 to -840.
y=-\frac{170}{7}
Divide both sides by 294.
18x-19\left(-\frac{170}{7}\right)=35
Substitute -\frac{170}{7} for y in 18x-19y=35. Because the resulting equation contains only one variable, you can solve for x directly.
18x+\frac{3230}{7}=35
Multiply -19 times -\frac{170}{7}.
18x=-\frac{2985}{7}
Subtract \frac{3230}{7} from both sides of the equation.
x=-\frac{995}{42}
Divide both sides by 18.
x=-\frac{995}{42},y=-\frac{170}{7}
The system is now solved.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
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Matrix
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Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
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