\left\{ \begin{array} { l } { \frac { x + 5 y } { 2 } + \frac { 11 x - 2 y } { 8 } = \frac { 2 x - 4 y + 6 } { 5 } } \\ { \frac { 2 x - 3 y } { 7 } - \frac { y - 2 x } { 5 } = \frac { 2 ( 9 x + 7 y ) } { 11 } } \end{array} \right.
Solve for x, y
x=-24
y=12
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20\left(x+5y\right)+5\left(11x-2y\right)=8\left(2x-4y+6\right)
Consider the first equation. Multiply both sides of the equation by 40, the least common multiple of 2,8,5.
20x+100y+5\left(11x-2y\right)=8\left(2x-4y+6\right)
Use the distributive property to multiply 20 by x+5y.
20x+100y+55x-10y=8\left(2x-4y+6\right)
Use the distributive property to multiply 5 by 11x-2y.
75x+100y-10y=8\left(2x-4y+6\right)
Combine 20x and 55x to get 75x.
75x+90y=8\left(2x-4y+6\right)
Combine 100y and -10y to get 90y.
75x+90y=16x-32y+48
Use the distributive property to multiply 8 by 2x-4y+6.
75x+90y-16x=-32y+48
Subtract 16x from both sides.
59x+90y=-32y+48
Combine 75x and -16x to get 59x.
59x+90y+32y=48
Add 32y to both sides.
59x+122y=48
Combine 90y and 32y to get 122y.
55\left(2x-3y\right)-77\left(y-2x\right)=35\times 2\left(9x+7y\right)
Consider the second equation. Multiply both sides of the equation by 385, the least common multiple of 7,5,11.
110x-165y-77\left(y-2x\right)=35\times 2\left(9x+7y\right)
Use the distributive property to multiply 55 by 2x-3y.
110x-165y-77y+154x=35\times 2\left(9x+7y\right)
Use the distributive property to multiply -77 by y-2x.
110x-242y+154x=35\times 2\left(9x+7y\right)
Combine -165y and -77y to get -242y.
264x-242y=35\times 2\left(9x+7y\right)
Combine 110x and 154x to get 264x.
264x-242y=70\left(9x+7y\right)
Multiply 35 and 2 to get 70.
264x-242y=630x+490y
Use the distributive property to multiply 70 by 9x+7y.
264x-242y-630x=490y
Subtract 630x from both sides.
-366x-242y=490y
Combine 264x and -630x to get -366x.
-366x-242y-490y=0
Subtract 490y from both sides.
-366x-732y=0
Combine -242y and -490y to get -732y.
59x+122y=48,-366x-732y=0
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.
59x+122y=48
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
59x=-122y+48
Subtract 122y from both sides of the equation.
x=\frac{1}{59}\left(-122y+48\right)
Divide both sides by 59.
x=-\frac{122}{59}y+\frac{48}{59}
Multiply \frac{1}{59} times -122y+48.
-366\left(-\frac{122}{59}y+\frac{48}{59}\right)-732y=0
Substitute \frac{-122y+48}{59} for x in the other equation, -366x-732y=0.
\frac{44652}{59}y-\frac{17568}{59}-732y=0
Multiply -366 times \frac{-122y+48}{59}.
\frac{1464}{59}y-\frac{17568}{59}=0
Add \frac{44652y}{59} to -732y.
\frac{1464}{59}y=\frac{17568}{59}
Add \frac{17568}{59} to both sides of the equation.
y=12
Divide both sides of the equation by \frac{1464}{59}, which is the same as multiplying both sides by the reciprocal of the fraction.
x=-\frac{122}{59}\times 12+\frac{48}{59}
Substitute 12 for y in x=-\frac{122}{59}y+\frac{48}{59}. Because the resulting equation contains only one variable, you can solve for x directly.
x=\frac{-1464+48}{59}
Multiply -\frac{122}{59} times 12.
x=-24
Add \frac{48}{59} to -\frac{1464}{59} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=-24,y=12
The system is now solved.
20\left(x+5y\right)+5\left(11x-2y\right)=8\left(2x-4y+6\right)
Consider the first equation. Multiply both sides of the equation by 40, the least common multiple of 2,8,5.
20x+100y+5\left(11x-2y\right)=8\left(2x-4y+6\right)
Use the distributive property to multiply 20 by x+5y.
20x+100y+55x-10y=8\left(2x-4y+6\right)
Use the distributive property to multiply 5 by 11x-2y.
75x+100y-10y=8\left(2x-4y+6\right)
Combine 20x and 55x to get 75x.
75x+90y=8\left(2x-4y+6\right)
Combine 100y and -10y to get 90y.
75x+90y=16x-32y+48
Use the distributive property to multiply 8 by 2x-4y+6.
75x+90y-16x=-32y+48
Subtract 16x from both sides.
59x+90y=-32y+48
Combine 75x and -16x to get 59x.
59x+90y+32y=48
Add 32y to both sides.
59x+122y=48
Combine 90y and 32y to get 122y.
55\left(2x-3y\right)-77\left(y-2x\right)=35\times 2\left(9x+7y\right)
Consider the second equation. Multiply both sides of the equation by 385, the least common multiple of 7,5,11.
110x-165y-77\left(y-2x\right)=35\times 2\left(9x+7y\right)
Use the distributive property to multiply 55 by 2x-3y.
110x-165y-77y+154x=35\times 2\left(9x+7y\right)
Use the distributive property to multiply -77 by y-2x.
110x-242y+154x=35\times 2\left(9x+7y\right)
Combine -165y and -77y to get -242y.
264x-242y=35\times 2\left(9x+7y\right)
Combine 110x and 154x to get 264x.
264x-242y=70\left(9x+7y\right)
Multiply 35 and 2 to get 70.
264x-242y=630x+490y
Use the distributive property to multiply 70 by 9x+7y.
264x-242y-630x=490y
Subtract 630x from both sides.
-366x-242y=490y
Combine 264x and -630x to get -366x.
-366x-242y-490y=0
Subtract 490y from both sides.
-366x-732y=0
Combine -242y and -490y to get -732y.
59x+122y=48,-366x-732y=0
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}59&122\\-366&-732\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}48\\0\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}59&122\\-366&-732\end{matrix}\right))\left(\begin{matrix}59&122\\-366&-732\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}59&122\\-366&-732\end{matrix}\right))\left(\begin{matrix}48\\0\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}59&122\\-366&-732\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}59&122\\-366&-732\end{matrix}\right))\left(\begin{matrix}48\\0\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}59&122\\-366&-732\end{matrix}\right))\left(\begin{matrix}48\\0\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{732}{59\left(-732\right)-122\left(-366\right)}&-\frac{122}{59\left(-732\right)-122\left(-366\right)}\\-\frac{-366}{59\left(-732\right)-122\left(-366\right)}&\frac{59}{59\left(-732\right)-122\left(-366\right)}\end{matrix}\right)\left(\begin{matrix}48\\0\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}{2}&-\frac{1}{12}\\\frac{1}{4}&\frac{59}{1464}\end{matrix}\right)\left(\begin{matrix}48\\0\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}\times 48\\\frac{1}{4}\times 48\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-24\\12\end{matrix}\right)
Do the arithmetic.
x=-24,y=12
Extract the matrix elements x and y.
20\left(x+5y\right)+5\left(11x-2y\right)=8\left(2x-4y+6\right)
Consider the first equation. Multiply both sides of the equation by 40, the least common multiple of 2,8,5.
20x+100y+5\left(11x-2y\right)=8\left(2x-4y+6\right)
Use the distributive property to multiply 20 by x+5y.
20x+100y+55x-10y=8\left(2x-4y+6\right)
Use the distributive property to multiply 5 by 11x-2y.
75x+100y-10y=8\left(2x-4y+6\right)
Combine 20x and 55x to get 75x.
75x+90y=8\left(2x-4y+6\right)
Combine 100y and -10y to get 90y.
75x+90y=16x-32y+48
Use the distributive property to multiply 8 by 2x-4y+6.
75x+90y-16x=-32y+48
Subtract 16x from both sides.
59x+90y=-32y+48
Combine 75x and -16x to get 59x.
59x+90y+32y=48
Add 32y to both sides.
59x+122y=48
Combine 90y and 32y to get 122y.
55\left(2x-3y\right)-77\left(y-2x\right)=35\times 2\left(9x+7y\right)
Consider the second equation. Multiply both sides of the equation by 385, the least common multiple of 7,5,11.
110x-165y-77\left(y-2x\right)=35\times 2\left(9x+7y\right)
Use the distributive property to multiply 55 by 2x-3y.
110x-165y-77y+154x=35\times 2\left(9x+7y\right)
Use the distributive property to multiply -77 by y-2x.
110x-242y+154x=35\times 2\left(9x+7y\right)
Combine -165y and -77y to get -242y.
264x-242y=35\times 2\left(9x+7y\right)
Combine 110x and 154x to get 264x.
264x-242y=70\left(9x+7y\right)
Multiply 35 and 2 to get 70.
264x-242y=630x+490y
Use the distributive property to multiply 70 by 9x+7y.
264x-242y-630x=490y
Subtract 630x from both sides.
-366x-242y=490y
Combine 264x and -630x to get -366x.
-366x-242y-490y=0
Subtract 490y from both sides.
-366x-732y=0
Combine -242y and -490y to get -732y.
59x+122y=48,-366x-732y=0
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.
-366\times 59x-366\times 122y=-366\times 48,59\left(-366\right)x+59\left(-732\right)y=0
To make 59x and -366x equal, multiply all terms on each side of the first equation by -366 and all terms on each side of the second by 59.
-21594x-44652y=-17568,-21594x-43188y=0
Simplify.
-21594x+21594x-44652y+43188y=-17568
Subtract -21594x-43188y=0 from -21594x-44652y=-17568 by subtracting like terms on each side of the equal sign.
-44652y+43188y=-17568
Add -21594x to 21594x. Terms -21594x and 21594x cancel out, leaving an equation with only one variable that can be solved.
-1464y=-17568
Add -44652y to 43188y.
y=12
Divide both sides by -1464.
-366x-732\times 12=0
Substitute 12 for y in -366x-732y=0. Because the resulting equation contains only one variable, you can solve for x directly.
-366x-8784=0
Multiply -732 times 12.
-366x=8784
Add 8784 to both sides of the equation.
x=-24
Divide both sides by -366.
x=-24,y=12
The system is now solved.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
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
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}