\left\{ \begin{array} { l } { \frac { x + 3 y } { 3 } = \frac { y + 1 } { 2 } + \frac { 3 x } { 4 } } \\ { \frac { 3 x + 5 y } { 5 } = \frac { x + 4 } { 2 } - \frac { x + y + 9 } { 10 } } \end{array} \right.
Solve for x, y
x=0
y=1
Graph
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4\left(x+3y\right)=6\left(y+1\right)+3\times 3x
Consider the first equation. Multiply both sides of the equation by 12, the least common multiple of 3,2,4.
4x+12y=6\left(y+1\right)+3\times 3x
Use the distributive property to multiply 4 by x+3y.
4x+12y=6y+6+3\times 3x
Use the distributive property to multiply 6 by y+1.
4x+12y=6y+6+9x
Multiply 3 and 3 to get 9.
4x+12y-6y=6+9x
Subtract 6y from both sides.
4x+6y=6+9x
Combine 12y and -6y to get 6y.
4x+6y-9x=6
Subtract 9x from both sides.
-5x+6y=6
Combine 4x and -9x to get -5x.
2\left(3x+5y\right)=5\left(x+4\right)-\left(x+y+9\right)
Consider the second equation. Multiply both sides of the equation by 10, the least common multiple of 5,2,10.
6x+10y=5\left(x+4\right)-\left(x+y+9\right)
Use the distributive property to multiply 2 by 3x+5y.
6x+10y=5x+20-\left(x+y+9\right)
Use the distributive property to multiply 5 by x+4.
6x+10y=5x+20-x-y-9
To find the opposite of x+y+9, find the opposite of each term.
6x+10y=4x+20-y-9
Combine 5x and -x to get 4x.
6x+10y=4x+11-y
Subtract 9 from 20 to get 11.
6x+10y-4x=11-y
Subtract 4x from both sides.
2x+10y=11-y
Combine 6x and -4x to get 2x.
2x+10y+y=11
Add y to both sides.
2x+11y=11
Combine 10y and y to get 11y.
-5x+6y=6,2x+11y=11
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.
-5x+6y=6
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
-5x=-6y+6
Subtract 6y from both sides of the equation.
x=-\frac{1}{5}\left(-6y+6\right)
Divide both sides by -5.
x=\frac{6}{5}y-\frac{6}{5}
Multiply -\frac{1}{5} times -6y+6.
2\left(\frac{6}{5}y-\frac{6}{5}\right)+11y=11
Substitute \frac{-6+6y}{5} for x in the other equation, 2x+11y=11.
\frac{12}{5}y-\frac{12}{5}+11y=11
Multiply 2 times \frac{-6+6y}{5}.
\frac{67}{5}y-\frac{12}{5}=11
Add \frac{12y}{5} to 11y.
\frac{67}{5}y=\frac{67}{5}
Add \frac{12}{5} to both sides of the equation.
y=1
Divide both sides of the equation by \frac{67}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.
x=\frac{6-6}{5}
Substitute 1 for y in x=\frac{6}{5}y-\frac{6}{5}. Because the resulting equation contains only one variable, you can solve for x directly.
x=0
Add -\frac{6}{5} to \frac{6}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=0,y=1
The system is now solved.
4\left(x+3y\right)=6\left(y+1\right)+3\times 3x
Consider the first equation. Multiply both sides of the equation by 12, the least common multiple of 3,2,4.
4x+12y=6\left(y+1\right)+3\times 3x
Use the distributive property to multiply 4 by x+3y.
4x+12y=6y+6+3\times 3x
Use the distributive property to multiply 6 by y+1.
4x+12y=6y+6+9x
Multiply 3 and 3 to get 9.
4x+12y-6y=6+9x
Subtract 6y from both sides.
4x+6y=6+9x
Combine 12y and -6y to get 6y.
4x+6y-9x=6
Subtract 9x from both sides.
-5x+6y=6
Combine 4x and -9x to get -5x.
2\left(3x+5y\right)=5\left(x+4\right)-\left(x+y+9\right)
Consider the second equation. Multiply both sides of the equation by 10, the least common multiple of 5,2,10.
6x+10y=5\left(x+4\right)-\left(x+y+9\right)
Use the distributive property to multiply 2 by 3x+5y.
6x+10y=5x+20-\left(x+y+9\right)
Use the distributive property to multiply 5 by x+4.
6x+10y=5x+20-x-y-9
To find the opposite of x+y+9, find the opposite of each term.
6x+10y=4x+20-y-9
Combine 5x and -x to get 4x.
6x+10y=4x+11-y
Subtract 9 from 20 to get 11.
6x+10y-4x=11-y
Subtract 4x from both sides.
2x+10y=11-y
Combine 6x and -4x to get 2x.
2x+10y+y=11
Add y to both sides.
2x+11y=11
Combine 10y and y to get 11y.
-5x+6y=6,2x+11y=11
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}-5&6\\2&11\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\11\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}-5&6\\2&11\end{matrix}\right))\left(\begin{matrix}-5&6\\2&11\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&6\\2&11\end{matrix}\right))\left(\begin{matrix}6\\11\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}-5&6\\2&11\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}-5&6\\2&11\end{matrix}\right))\left(\begin{matrix}6\\11\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}-5&6\\2&11\end{matrix}\right))\left(\begin{matrix}6\\11\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{11}{-5\times 11-6\times 2}&-\frac{6}{-5\times 11-6\times 2}\\-\frac{2}{-5\times 11-6\times 2}&-\frac{5}{-5\times 11-6\times 2}\end{matrix}\right)\left(\begin{matrix}6\\11\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{11}{67}&\frac{6}{67}\\\frac{2}{67}&\frac{5}{67}\end{matrix}\right)\left(\begin{matrix}6\\11\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{11}{67}\times 6+\frac{6}{67}\times 11\\\frac{2}{67}\times 6+\frac{5}{67}\times 11\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\1\end{matrix}\right)
Do the arithmetic.
x=0,y=1
Extract the matrix elements x and y.
4\left(x+3y\right)=6\left(y+1\right)+3\times 3x
Consider the first equation. Multiply both sides of the equation by 12, the least common multiple of 3,2,4.
4x+12y=6\left(y+1\right)+3\times 3x
Use the distributive property to multiply 4 by x+3y.
4x+12y=6y+6+3\times 3x
Use the distributive property to multiply 6 by y+1.
4x+12y=6y+6+9x
Multiply 3 and 3 to get 9.
4x+12y-6y=6+9x
Subtract 6y from both sides.
4x+6y=6+9x
Combine 12y and -6y to get 6y.
4x+6y-9x=6
Subtract 9x from both sides.
-5x+6y=6
Combine 4x and -9x to get -5x.
2\left(3x+5y\right)=5\left(x+4\right)-\left(x+y+9\right)
Consider the second equation. Multiply both sides of the equation by 10, the least common multiple of 5,2,10.
6x+10y=5\left(x+4\right)-\left(x+y+9\right)
Use the distributive property to multiply 2 by 3x+5y.
6x+10y=5x+20-\left(x+y+9\right)
Use the distributive property to multiply 5 by x+4.
6x+10y=5x+20-x-y-9
To find the opposite of x+y+9, find the opposite of each term.
6x+10y=4x+20-y-9
Combine 5x and -x to get 4x.
6x+10y=4x+11-y
Subtract 9 from 20 to get 11.
6x+10y-4x=11-y
Subtract 4x from both sides.
2x+10y=11-y
Combine 6x and -4x to get 2x.
2x+10y+y=11
Add y to both sides.
2x+11y=11
Combine 10y and y to get 11y.
-5x+6y=6,2x+11y=11
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.
2\left(-5\right)x+2\times 6y=2\times 6,-5\times 2x-5\times 11y=-5\times 11
To make -5x and 2x equal, multiply all terms on each side of the first equation by 2 and all terms on each side of the second by -5.
-10x+12y=12,-10x-55y=-55
Simplify.
-10x+10x+12y+55y=12+55
Subtract -10x-55y=-55 from -10x+12y=12 by subtracting like terms on each side of the equal sign.
12y+55y=12+55
Add -10x to 10x. Terms -10x and 10x cancel out, leaving an equation with only one variable that can be solved.
67y=12+55
Add 12y to 55y.
67y=67
Add 12 to 55.
y=1
Divide both sides by 67.
2x+11=11
Substitute 1 for y in 2x+11y=11. Because the resulting equation contains only one variable, you can solve for x directly.
2x=0
Subtract 11 from both sides of the equation.
x=0
Divide both sides by 2.
x=0,y=1
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}