\left\{ \begin{array} { l } { \frac { 3 x - 2 y } { 2 } + \frac { 2 x - 3 y } { 3 } = \frac { 1 } { 6 } } \\ { \frac { 3 x - 2 y } { 3 } + \frac { 2 x - 3 y } { 2 } = - \frac { 1 } { 6 } } \end{array} \right.
Solve for x, y
x=1
y=1
Graph
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3\left(3x-2y\right)+2\left(2x-3y\right)=1
Consider the first equation. Multiply both sides of the equation by 6, the least common multiple of 2,3,6.
9x-6y+2\left(2x-3y\right)=1
Use the distributive property to multiply 3 by 3x-2y.
9x-6y+4x-6y=1
Use the distributive property to multiply 2 by 2x-3y.
13x-6y-6y=1
Combine 9x and 4x to get 13x.
13x-12y=1
Combine -6y and -6y to get -12y.
2\left(3x-2y\right)+3\left(2x-3y\right)=-1
Consider the second equation. Multiply both sides of the equation by 6, the least common multiple of 3,2,6.
6x-4y+3\left(2x-3y\right)=-1
Use the distributive property to multiply 2 by 3x-2y.
6x-4y+6x-9y=-1
Use the distributive property to multiply 3 by 2x-3y.
12x-4y-9y=-1
Combine 6x and 6x to get 12x.
12x-13y=-1
Combine -4y and -9y to get -13y.
13x-12y=1,12x-13y=-1
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.
13x-12y=1
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
13x=12y+1
Add 12y to both sides of the equation.
x=\frac{1}{13}\left(12y+1\right)
Divide both sides by 13.
x=\frac{12}{13}y+\frac{1}{13}
Multiply \frac{1}{13} times 12y+1.
12\left(\frac{12}{13}y+\frac{1}{13}\right)-13y=-1
Substitute \frac{12y+1}{13} for x in the other equation, 12x-13y=-1.
\frac{144}{13}y+\frac{12}{13}-13y=-1
Multiply 12 times \frac{12y+1}{13}.
-\frac{25}{13}y+\frac{12}{13}=-1
Add \frac{144y}{13} to -13y.
-\frac{25}{13}y=-\frac{25}{13}
Subtract \frac{12}{13} from both sides of the equation.
y=1
Divide both sides of the equation by -\frac{25}{13}, which is the same as multiplying both sides by the reciprocal of the fraction.
x=\frac{12+1}{13}
Substitute 1 for y in x=\frac{12}{13}y+\frac{1}{13}. Because the resulting equation contains only one variable, you can solve for x directly.
x=1
Add \frac{1}{13} to \frac{12}{13} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=1,y=1
The system is now solved.
3\left(3x-2y\right)+2\left(2x-3y\right)=1
Consider the first equation. Multiply both sides of the equation by 6, the least common multiple of 2,3,6.
9x-6y+2\left(2x-3y\right)=1
Use the distributive property to multiply 3 by 3x-2y.
9x-6y+4x-6y=1
Use the distributive property to multiply 2 by 2x-3y.
13x-6y-6y=1
Combine 9x and 4x to get 13x.
13x-12y=1
Combine -6y and -6y to get -12y.
2\left(3x-2y\right)+3\left(2x-3y\right)=-1
Consider the second equation. Multiply both sides of the equation by 6, the least common multiple of 3,2,6.
6x-4y+3\left(2x-3y\right)=-1
Use the distributive property to multiply 2 by 3x-2y.
6x-4y+6x-9y=-1
Use the distributive property to multiply 3 by 2x-3y.
12x-4y-9y=-1
Combine 6x and 6x to get 12x.
12x-13y=-1
Combine -4y and -9y to get -13y.
13x-12y=1,12x-13y=-1
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}13&-12\\12&-13\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\-1\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}13&-12\\12&-13\end{matrix}\right))\left(\begin{matrix}13&-12\\12&-13\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}13&-12\\12&-13\end{matrix}\right))\left(\begin{matrix}1\\-1\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}13&-12\\12&-13\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}13&-12\\12&-13\end{matrix}\right))\left(\begin{matrix}1\\-1\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}13&-12\\12&-13\end{matrix}\right))\left(\begin{matrix}1\\-1\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{13}{13\left(-13\right)-\left(-12\times 12\right)}&-\frac{-12}{13\left(-13\right)-\left(-12\times 12\right)}\\-\frac{12}{13\left(-13\right)-\left(-12\times 12\right)}&\frac{13}{13\left(-13\right)-\left(-12\times 12\right)}\end{matrix}\right)\left(\begin{matrix}1\\-1\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{13}{25}&-\frac{12}{25}\\\frac{12}{25}&-\frac{13}{25}\end{matrix}\right)\left(\begin{matrix}1\\-1\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{25}-\frac{12}{25}\left(-1\right)\\\frac{12}{25}-\frac{13}{25}\left(-1\right)\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\1\end{matrix}\right)
Do the arithmetic.
x=1,y=1
Extract the matrix elements x and y.
3\left(3x-2y\right)+2\left(2x-3y\right)=1
Consider the first equation. Multiply both sides of the equation by 6, the least common multiple of 2,3,6.
9x-6y+2\left(2x-3y\right)=1
Use the distributive property to multiply 3 by 3x-2y.
9x-6y+4x-6y=1
Use the distributive property to multiply 2 by 2x-3y.
13x-6y-6y=1
Combine 9x and 4x to get 13x.
13x-12y=1
Combine -6y and -6y to get -12y.
2\left(3x-2y\right)+3\left(2x-3y\right)=-1
Consider the second equation. Multiply both sides of the equation by 6, the least common multiple of 3,2,6.
6x-4y+3\left(2x-3y\right)=-1
Use the distributive property to multiply 2 by 3x-2y.
6x-4y+6x-9y=-1
Use the distributive property to multiply 3 by 2x-3y.
12x-4y-9y=-1
Combine 6x and 6x to get 12x.
12x-13y=-1
Combine -4y and -9y to get -13y.
13x-12y=1,12x-13y=-1
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.
12\times 13x+12\left(-12\right)y=12,13\times 12x+13\left(-13\right)y=13\left(-1\right)
To make 13x and 12x equal, multiply all terms on each side of the first equation by 12 and all terms on each side of the second by 13.
156x-144y=12,156x-169y=-13
Simplify.
156x-156x-144y+169y=12+13
Subtract 156x-169y=-13 from 156x-144y=12 by subtracting like terms on each side of the equal sign.
-144y+169y=12+13
Add 156x to -156x. Terms 156x and -156x cancel out, leaving an equation with only one variable that can be solved.
25y=12+13
Add -144y to 169y.
25y=25
Add 12 to 13.
y=1
Divide both sides by 25.
12x-13=-1
Substitute 1 for y in 12x-13y=-1. Because the resulting equation contains only one variable, you can solve for x directly.
12x=12
Add 13 to both sides of the equation.
x=1
Divide both sides by 12.
x=1,y=1
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}