\left\{ \begin{array} { l } { \frac { x } { 3 } + \frac { y } { 4 } = \frac { 2 } { 2 } - \frac { 6 } { 6 } } \\ { \frac { 2 x + y } { 5 } - \frac { y - 2 } { 2 } = \frac { x + y - 3 } { 4 } - \frac { y - x - 1 } { 10 } } \end{array} \right.
Solve for x, y
x = -\frac{33}{13} = -2\frac{7}{13} \approx -2.538461538
y = \frac{44}{13} = 3\frac{5}{13} \approx 3.384615385
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4x+3y=6\times 2-2\times 6
Consider the first equation. Multiply both sides of the equation by 12, the least common multiple of 3,4,2,6.
4x+3y=12-12
Do the multiplications.
4x+3y=0
Subtract 12 from 12 to get 0.
4\left(2x+y\right)-10\left(y-2\right)=5\left(x+y-3\right)-2\left(y-x-1\right)
Consider the second equation. Multiply both sides of the equation by 20, the least common multiple of 5,2,4,10.
8x+4y-10\left(y-2\right)=5\left(x+y-3\right)-2\left(y-x-1\right)
Use the distributive property to multiply 4 by 2x+y.
8x+4y-10y+20=5\left(x+y-3\right)-2\left(y-x-1\right)
Use the distributive property to multiply -10 by y-2.
8x-6y+20=5\left(x+y-3\right)-2\left(y-x-1\right)
Combine 4y and -10y to get -6y.
8x-6y+20=5x+5y-15-2\left(y-x-1\right)
Use the distributive property to multiply 5 by x+y-3.
8x-6y+20=5x+5y-15-2y+2x+2
Use the distributive property to multiply -2 by y-x-1.
8x-6y+20=5x+3y-15+2x+2
Combine 5y and -2y to get 3y.
8x-6y+20=7x+3y-15+2
Combine 5x and 2x to get 7x.
8x-6y+20=7x+3y-13
Add -15 and 2 to get -13.
8x-6y+20-7x=3y-13
Subtract 7x from both sides.
x-6y+20=3y-13
Combine 8x and -7x to get x.
x-6y+20-3y=-13
Subtract 3y from both sides.
x-9y+20=-13
Combine -6y and -3y to get -9y.
x-9y=-13-20
Subtract 20 from both sides.
x-9y=-33
Subtract 20 from -13 to get -33.
4x+3y=0,x-9y=-33
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.
4x+3y=0
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
4x=-3y
Subtract 3y from both sides of the equation.
x=\frac{1}{4}\left(-3\right)y
Divide both sides by 4.
x=-\frac{3}{4}y
Multiply \frac{1}{4} times -3y.
-\frac{3}{4}y-9y=-33
Substitute -\frac{3y}{4} for x in the other equation, x-9y=-33.
-\frac{39}{4}y=-33
Add -\frac{3y}{4} to -9y.
y=\frac{44}{13}
Divide both sides of the equation by -\frac{39}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.
x=-\frac{3}{4}\times \frac{44}{13}
Substitute \frac{44}{13} for y in x=-\frac{3}{4}y. Because the resulting equation contains only one variable, you can solve for x directly.
x=-\frac{33}{13}
Multiply -\frac{3}{4} times \frac{44}{13} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
x=-\frac{33}{13},y=\frac{44}{13}
The system is now solved.
4x+3y=6\times 2-2\times 6
Consider the first equation. Multiply both sides of the equation by 12, the least common multiple of 3,4,2,6.
4x+3y=12-12
Do the multiplications.
4x+3y=0
Subtract 12 from 12 to get 0.
4\left(2x+y\right)-10\left(y-2\right)=5\left(x+y-3\right)-2\left(y-x-1\right)
Consider the second equation. Multiply both sides of the equation by 20, the least common multiple of 5,2,4,10.
8x+4y-10\left(y-2\right)=5\left(x+y-3\right)-2\left(y-x-1\right)
Use the distributive property to multiply 4 by 2x+y.
8x+4y-10y+20=5\left(x+y-3\right)-2\left(y-x-1\right)
Use the distributive property to multiply -10 by y-2.
8x-6y+20=5\left(x+y-3\right)-2\left(y-x-1\right)
Combine 4y and -10y to get -6y.
8x-6y+20=5x+5y-15-2\left(y-x-1\right)
Use the distributive property to multiply 5 by x+y-3.
8x-6y+20=5x+5y-15-2y+2x+2
Use the distributive property to multiply -2 by y-x-1.
8x-6y+20=5x+3y-15+2x+2
Combine 5y and -2y to get 3y.
8x-6y+20=7x+3y-15+2
Combine 5x and 2x to get 7x.
8x-6y+20=7x+3y-13
Add -15 and 2 to get -13.
8x-6y+20-7x=3y-13
Subtract 7x from both sides.
x-6y+20=3y-13
Combine 8x and -7x to get x.
x-6y+20-3y=-13
Subtract 3y from both sides.
x-9y+20=-13
Combine -6y and -3y to get -9y.
x-9y=-13-20
Subtract 20 from both sides.
x-9y=-33
Subtract 20 from -13 to get -33.
4x+3y=0,x-9y=-33
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}4&3\\1&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\-33\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}4&3\\1&-9\end{matrix}\right))\left(\begin{matrix}4&3\\1&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\1&-9\end{matrix}\right))\left(\begin{matrix}0\\-33\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}4&3\\1&-9\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}4&3\\1&-9\end{matrix}\right))\left(\begin{matrix}0\\-33\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}4&3\\1&-9\end{matrix}\right))\left(\begin{matrix}0\\-33\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{9}{4\left(-9\right)-3}&-\frac{3}{4\left(-9\right)-3}\\-\frac{1}{4\left(-9\right)-3}&\frac{4}{4\left(-9\right)-3}\end{matrix}\right)\left(\begin{matrix}0\\-33\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{3}{13}&\frac{1}{13}\\\frac{1}{39}&-\frac{4}{39}\end{matrix}\right)\left(\begin{matrix}0\\-33\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{13}\left(-33\right)\\-\frac{4}{39}\left(-33\right)\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{33}{13}\\\frac{44}{13}\end{matrix}\right)
Do the arithmetic.
x=-\frac{33}{13},y=\frac{44}{13}
Extract the matrix elements x and y.
4x+3y=6\times 2-2\times 6
Consider the first equation. Multiply both sides of the equation by 12, the least common multiple of 3,4,2,6.
4x+3y=12-12
Do the multiplications.
4x+3y=0
Subtract 12 from 12 to get 0.
4\left(2x+y\right)-10\left(y-2\right)=5\left(x+y-3\right)-2\left(y-x-1\right)
Consider the second equation. Multiply both sides of the equation by 20, the least common multiple of 5,2,4,10.
8x+4y-10\left(y-2\right)=5\left(x+y-3\right)-2\left(y-x-1\right)
Use the distributive property to multiply 4 by 2x+y.
8x+4y-10y+20=5\left(x+y-3\right)-2\left(y-x-1\right)
Use the distributive property to multiply -10 by y-2.
8x-6y+20=5\left(x+y-3\right)-2\left(y-x-1\right)
Combine 4y and -10y to get -6y.
8x-6y+20=5x+5y-15-2\left(y-x-1\right)
Use the distributive property to multiply 5 by x+y-3.
8x-6y+20=5x+5y-15-2y+2x+2
Use the distributive property to multiply -2 by y-x-1.
8x-6y+20=5x+3y-15+2x+2
Combine 5y and -2y to get 3y.
8x-6y+20=7x+3y-15+2
Combine 5x and 2x to get 7x.
8x-6y+20=7x+3y-13
Add -15 and 2 to get -13.
8x-6y+20-7x=3y-13
Subtract 7x from both sides.
x-6y+20=3y-13
Combine 8x and -7x to get x.
x-6y+20-3y=-13
Subtract 3y from both sides.
x-9y+20=-13
Combine -6y and -3y to get -9y.
x-9y=-13-20
Subtract 20 from both sides.
x-9y=-33
Subtract 20 from -13 to get -33.
4x+3y=0,x-9y=-33
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.
4x+3y=0,4x+4\left(-9\right)y=4\left(-33\right)
To make 4x and x equal, multiply all terms on each side of the first equation by 1 and all terms on each side of the second by 4.
4x+3y=0,4x-36y=-132
Simplify.
4x-4x+3y+36y=132
Subtract 4x-36y=-132 from 4x+3y=0 by subtracting like terms on each side of the equal sign.
3y+36y=132
Add 4x to -4x. Terms 4x and -4x cancel out, leaving an equation with only one variable that can be solved.
39y=132
Add 3y to 36y.
y=\frac{44}{13}
Divide both sides by 39.
x-9\times \frac{44}{13}=-33
Substitute \frac{44}{13} for y in x-9y=-33. Because the resulting equation contains only one variable, you can solve for x directly.
x-\frac{396}{13}=-33
Multiply -9 times \frac{44}{13}.
x=-\frac{33}{13}
Add \frac{396}{13} to both sides of the equation.
x=-\frac{33}{13},y=\frac{44}{13}
The system is now solved.
Examples
Quadratic equation
{ 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}