\left\{ \begin{array} { l } { \frac { y } { y - 5 } - \frac { x } { x + 1 } = \frac { 2 x - 3 y + 7 } { x y - 5 x + y - 5 } } \\ { \frac { x - 2 } { 3 } + \frac { y } { 2 } = \frac { 1 } { 6 } } \end{array} \right.
Solve for y, x
x=1
y=1
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\left(x+1\right)y-\left(y-5\right)x=2x-3y+7
Consider the first equation. Variable x cannot be equal to -1 since division by zero is not defined. Variable y cannot be equal to 5 since division by zero is not defined. Multiply both sides of the equation by \left(y-5\right)\left(x+1\right), the least common multiple of y-5,x+1,xy-5x+y-5.
xy+y-\left(y-5\right)x=2x-3y+7
Use the distributive property to multiply x+1 by y.
xy+y-\left(yx-5x\right)=2x-3y+7
Use the distributive property to multiply y-5 by x.
xy+y-yx+5x=2x-3y+7
To find the opposite of yx-5x, find the opposite of each term.
y+5x=2x-3y+7
Combine xy and -yx to get 0.
y+5x-2x=-3y+7
Subtract 2x from both sides.
y+3x=-3y+7
Combine 5x and -2x to get 3x.
y+3x+3y=7
Add 3y to both sides.
4y+3x=7
Combine y and 3y to get 4y.
2\left(x-2\right)+3y=1
Consider the second equation. Multiply both sides of the equation by 6, the least common multiple of 3,2,6.
2x-4+3y=1
Use the distributive property to multiply 2 by x-2.
2x+3y=1+4
Add 4 to both sides.
2x+3y=5
Add 1 and 4 to get 5.
4y+3x=7,3y+2x=5
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.
4y+3x=7
Choose one of the equations and solve it for y by isolating y on the left hand side of the equal sign.
4y=-3x+7
Subtract 3x from both sides of the equation.
y=\frac{1}{4}\left(-3x+7\right)
Divide both sides by 4.
y=-\frac{3}{4}x+\frac{7}{4}
Multiply \frac{1}{4} times -3x+7.
3\left(-\frac{3}{4}x+\frac{7}{4}\right)+2x=5
Substitute \frac{-3x+7}{4} for y in the other equation, 3y+2x=5.
-\frac{9}{4}x+\frac{21}{4}+2x=5
Multiply 3 times \frac{-3x+7}{4}.
-\frac{1}{4}x+\frac{21}{4}=5
Add -\frac{9x}{4} to 2x.
-\frac{1}{4}x=-\frac{1}{4}
Subtract \frac{21}{4} from both sides of the equation.
x=1
Multiply both sides by -4.
y=\frac{-3+7}{4}
Substitute 1 for x in y=-\frac{3}{4}x+\frac{7}{4}. Because the resulting equation contains only one variable, you can solve for y directly.
y=1
Add \frac{7}{4} to -\frac{3}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
y=1,x=1
The system is now solved.
\left(x+1\right)y-\left(y-5\right)x=2x-3y+7
Consider the first equation. Variable x cannot be equal to -1 since division by zero is not defined. Variable y cannot be equal to 5 since division by zero is not defined. Multiply both sides of the equation by \left(y-5\right)\left(x+1\right), the least common multiple of y-5,x+1,xy-5x+y-5.
xy+y-\left(y-5\right)x=2x-3y+7
Use the distributive property to multiply x+1 by y.
xy+y-\left(yx-5x\right)=2x-3y+7
Use the distributive property to multiply y-5 by x.
xy+y-yx+5x=2x-3y+7
To find the opposite of yx-5x, find the opposite of each term.
y+5x=2x-3y+7
Combine xy and -yx to get 0.
y+5x-2x=-3y+7
Subtract 2x from both sides.
y+3x=-3y+7
Combine 5x and -2x to get 3x.
y+3x+3y=7
Add 3y to both sides.
4y+3x=7
Combine y and 3y to get 4y.
2\left(x-2\right)+3y=1
Consider the second equation. Multiply both sides of the equation by 6, the least common multiple of 3,2,6.
2x-4+3y=1
Use the distributive property to multiply 2 by x-2.
2x+3y=1+4
Add 4 to both sides.
2x+3y=5
Add 1 and 4 to get 5.
4y+3x=7,3y+2x=5
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}4&3\\3&2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}7\\5\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}4&3\\3&2\end{matrix}\right))\left(\begin{matrix}4&3\\3&2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\3&2\end{matrix}\right))\left(\begin{matrix}7\\5\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}4&3\\3&2\end{matrix}\right).
\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\3&2\end{matrix}\right))\left(\begin{matrix}7\\5\end{matrix}\right)
The product of a matrix and its inverse is the identity matrix.
\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\3&2\end{matrix}\right))\left(\begin{matrix}7\\5\end{matrix}\right)
Multiply the matrices on the left hand side of the equal sign.
\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{4\times 2-3\times 3}&-\frac{3}{4\times 2-3\times 3}\\-\frac{3}{4\times 2-3\times 3}&\frac{4}{4\times 2-3\times 3}\end{matrix}\right)\left(\begin{matrix}7\\5\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}y\\x\end{matrix}\right)=\left(\begin{matrix}-2&3\\3&-4\end{matrix}\right)\left(\begin{matrix}7\\5\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-2\times 7+3\times 5\\3\times 7-4\times 5\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}1\\1\end{matrix}\right)
Do the arithmetic.
y=1,x=1
Extract the matrix elements y and x.
\left(x+1\right)y-\left(y-5\right)x=2x-3y+7
Consider the first equation. Variable x cannot be equal to -1 since division by zero is not defined. Variable y cannot be equal to 5 since division by zero is not defined. Multiply both sides of the equation by \left(y-5\right)\left(x+1\right), the least common multiple of y-5,x+1,xy-5x+y-5.
xy+y-\left(y-5\right)x=2x-3y+7
Use the distributive property to multiply x+1 by y.
xy+y-\left(yx-5x\right)=2x-3y+7
Use the distributive property to multiply y-5 by x.
xy+y-yx+5x=2x-3y+7
To find the opposite of yx-5x, find the opposite of each term.
y+5x=2x-3y+7
Combine xy and -yx to get 0.
y+5x-2x=-3y+7
Subtract 2x from both sides.
y+3x=-3y+7
Combine 5x and -2x to get 3x.
y+3x+3y=7
Add 3y to both sides.
4y+3x=7
Combine y and 3y to get 4y.
2\left(x-2\right)+3y=1
Consider the second equation. Multiply both sides of the equation by 6, the least common multiple of 3,2,6.
2x-4+3y=1
Use the distributive property to multiply 2 by x-2.
2x+3y=1+4
Add 4 to both sides.
2x+3y=5
Add 1 and 4 to get 5.
4y+3x=7,3y+2x=5
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 4y+3\times 3x=3\times 7,4\times 3y+4\times 2x=4\times 5
To make 4y and 3y equal, multiply all terms on each side of the first equation by 3 and all terms on each side of the second by 4.
12y+9x=21,12y+8x=20
Simplify.
12y-12y+9x-8x=21-20
Subtract 12y+8x=20 from 12y+9x=21 by subtracting like terms on each side of the equal sign.
9x-8x=21-20
Add 12y to -12y. Terms 12y and -12y cancel out, leaving an equation with only one variable that can be solved.
x=21-20
Add 9x to -8x.
x=1
Add 21 to -20.
3y+2=5
Substitute 1 for x in 3y+2x=5. Because the resulting equation contains only one variable, you can solve for y directly.
3y=3
Subtract 2 from both sides of the equation.
y=1
Divide both sides by 3.
y=1,x=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}