\left\{ \begin{array} { l } { \frac { 4 x + 3 y } { 5 } - \frac { x - 6 } { 3 } = 5 - \frac { y + 2 } { 10 } } \\ { \frac { x + 6 } { 4 } + \frac { 2 - y } { 6 } = 2 - \frac { 6 y - 13 } { 12 } } \end{array} \right.
Solve for x, y
x=-3
y=6
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6\left(4x+3y\right)-10\left(x-6\right)=150-3\left(y+2\right)
Consider the first equation. Multiply both sides of the equation by 30, the least common multiple of 5,3,10.
24x+18y-10\left(x-6\right)=150-3\left(y+2\right)
Use the distributive property to multiply 6 by 4x+3y.
24x+18y-10x+60=150-3\left(y+2\right)
Use the distributive property to multiply -10 by x-6.
14x+18y+60=150-3\left(y+2\right)
Combine 24x and -10x to get 14x.
14x+18y+60=150-3y-6
Use the distributive property to multiply -3 by y+2.
14x+18y+60=144-3y
Subtract 6 from 150 to get 144.
14x+18y+60+3y=144
Add 3y to both sides.
14x+21y+60=144
Combine 18y and 3y to get 21y.
14x+21y=144-60
Subtract 60 from both sides.
14x+21y=84
Subtract 60 from 144 to get 84.
3\left(x+6\right)+2\left(2-y\right)=24-\left(6y-13\right)
Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 4,6,12.
3x+18+2\left(2-y\right)=24-\left(6y-13\right)
Use the distributive property to multiply 3 by x+6.
3x+18+4-2y=24-\left(6y-13\right)
Use the distributive property to multiply 2 by 2-y.
3x+22-2y=24-\left(6y-13\right)
Add 18 and 4 to get 22.
3x+22-2y=24-6y+13
To find the opposite of 6y-13, find the opposite of each term.
3x+22-2y=37-6y
Add 24 and 13 to get 37.
3x+22-2y+6y=37
Add 6y to both sides.
3x+22+4y=37
Combine -2y and 6y to get 4y.
3x+4y=37-22
Subtract 22 from both sides.
3x+4y=15
Subtract 22 from 37 to get 15.
14x+21y=84,3x+4y=15
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.
14x+21y=84
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
14x=-21y+84
Subtract 21y from both sides of the equation.
x=\frac{1}{14}\left(-21y+84\right)
Divide both sides by 14.
x=-\frac{3}{2}y+6
Multiply \frac{1}{14} times -21y+84.
3\left(-\frac{3}{2}y+6\right)+4y=15
Substitute -\frac{3y}{2}+6 for x in the other equation, 3x+4y=15.
-\frac{9}{2}y+18+4y=15
Multiply 3 times -\frac{3y}{2}+6.
-\frac{1}{2}y+18=15
Add -\frac{9y}{2} to 4y.
-\frac{1}{2}y=-3
Subtract 18 from both sides of the equation.
y=6
Multiply both sides by -2.
x=-\frac{3}{2}\times 6+6
Substitute 6 for y in x=-\frac{3}{2}y+6. Because the resulting equation contains only one variable, you can solve for x directly.
x=-9+6
Multiply -\frac{3}{2} times 6.
x=-3
Add 6 to -9.
x=-3,y=6
The system is now solved.
6\left(4x+3y\right)-10\left(x-6\right)=150-3\left(y+2\right)
Consider the first equation. Multiply both sides of the equation by 30, the least common multiple of 5,3,10.
24x+18y-10\left(x-6\right)=150-3\left(y+2\right)
Use the distributive property to multiply 6 by 4x+3y.
24x+18y-10x+60=150-3\left(y+2\right)
Use the distributive property to multiply -10 by x-6.
14x+18y+60=150-3\left(y+2\right)
Combine 24x and -10x to get 14x.
14x+18y+60=150-3y-6
Use the distributive property to multiply -3 by y+2.
14x+18y+60=144-3y
Subtract 6 from 150 to get 144.
14x+18y+60+3y=144
Add 3y to both sides.
14x+21y+60=144
Combine 18y and 3y to get 21y.
14x+21y=144-60
Subtract 60 from both sides.
14x+21y=84
Subtract 60 from 144 to get 84.
3\left(x+6\right)+2\left(2-y\right)=24-\left(6y-13\right)
Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 4,6,12.
3x+18+2\left(2-y\right)=24-\left(6y-13\right)
Use the distributive property to multiply 3 by x+6.
3x+18+4-2y=24-\left(6y-13\right)
Use the distributive property to multiply 2 by 2-y.
3x+22-2y=24-\left(6y-13\right)
Add 18 and 4 to get 22.
3x+22-2y=24-6y+13
To find the opposite of 6y-13, find the opposite of each term.
3x+22-2y=37-6y
Add 24 and 13 to get 37.
3x+22-2y+6y=37
Add 6y to both sides.
3x+22+4y=37
Combine -2y and 6y to get 4y.
3x+4y=37-22
Subtract 22 from both sides.
3x+4y=15
Subtract 22 from 37 to get 15.
14x+21y=84,3x+4y=15
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}14&21\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}84\\15\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}14&21\\3&4\end{matrix}\right))\left(\begin{matrix}14&21\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}14&21\\3&4\end{matrix}\right))\left(\begin{matrix}84\\15\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}14&21\\3&4\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}14&21\\3&4\end{matrix}\right))\left(\begin{matrix}84\\15\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}14&21\\3&4\end{matrix}\right))\left(\begin{matrix}84\\15\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{4}{14\times 4-21\times 3}&-\frac{21}{14\times 4-21\times 3}\\-\frac{3}{14\times 4-21\times 3}&\frac{14}{14\times 4-21\times 3}\end{matrix}\right)\left(\begin{matrix}84\\15\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{4}{7}&3\\\frac{3}{7}&-2\end{matrix}\right)\left(\begin{matrix}84\\15\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{7}\times 84+3\times 15\\\frac{3}{7}\times 84-2\times 15\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\6\end{matrix}\right)
Do the arithmetic.
x=-3,y=6
Extract the matrix elements x and y.
6\left(4x+3y\right)-10\left(x-6\right)=150-3\left(y+2\right)
Consider the first equation. Multiply both sides of the equation by 30, the least common multiple of 5,3,10.
24x+18y-10\left(x-6\right)=150-3\left(y+2\right)
Use the distributive property to multiply 6 by 4x+3y.
24x+18y-10x+60=150-3\left(y+2\right)
Use the distributive property to multiply -10 by x-6.
14x+18y+60=150-3\left(y+2\right)
Combine 24x and -10x to get 14x.
14x+18y+60=150-3y-6
Use the distributive property to multiply -3 by y+2.
14x+18y+60=144-3y
Subtract 6 from 150 to get 144.
14x+18y+60+3y=144
Add 3y to both sides.
14x+21y+60=144
Combine 18y and 3y to get 21y.
14x+21y=144-60
Subtract 60 from both sides.
14x+21y=84
Subtract 60 from 144 to get 84.
3\left(x+6\right)+2\left(2-y\right)=24-\left(6y-13\right)
Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 4,6,12.
3x+18+2\left(2-y\right)=24-\left(6y-13\right)
Use the distributive property to multiply 3 by x+6.
3x+18+4-2y=24-\left(6y-13\right)
Use the distributive property to multiply 2 by 2-y.
3x+22-2y=24-\left(6y-13\right)
Add 18 and 4 to get 22.
3x+22-2y=24-6y+13
To find the opposite of 6y-13, find the opposite of each term.
3x+22-2y=37-6y
Add 24 and 13 to get 37.
3x+22-2y+6y=37
Add 6y to both sides.
3x+22+4y=37
Combine -2y and 6y to get 4y.
3x+4y=37-22
Subtract 22 from both sides.
3x+4y=15
Subtract 22 from 37 to get 15.
14x+21y=84,3x+4y=15
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.
3\times 14x+3\times 21y=3\times 84,14\times 3x+14\times 4y=14\times 15
To make 14x and 3x equal, multiply all terms on each side of the first equation by 3 and all terms on each side of the second by 14.
42x+63y=252,42x+56y=210
Simplify.
42x-42x+63y-56y=252-210
Subtract 42x+56y=210 from 42x+63y=252 by subtracting like terms on each side of the equal sign.
63y-56y=252-210
Add 42x to -42x. Terms 42x and -42x cancel out, leaving an equation with only one variable that can be solved.
7y=252-210
Add 63y to -56y.
7y=42
Add 252 to -210.
y=6
Divide both sides by 7.
3x+4\times 6=15
Substitute 6 for y in 3x+4y=15. Because the resulting equation contains only one variable, you can solve for x directly.
3x+24=15
Multiply 4 times 6.
3x=-9
Subtract 24 from both sides of the equation.
x=-3
Divide both sides by 3.
x=-3,y=6
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}