\left\{ \begin{array} { c } { \frac { 3 x - 2 y } { 5 } - x = \frac { 3 y - 5 x } { 3 } + 1 } \\ { \frac { 1 } { 3 } ( 2 x - 3 y ) + 5 = \frac { 4 x - 3 y } { 2 } + y } \end{array} \right.
Solve for x, y
x=3
y=2
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3\left(3x-2y\right)-15x=5\left(3y-5x\right)+15
Consider the first equation. Multiply both sides of the equation by 15, the least common multiple of 5,3.
9x-6y-15x=5\left(3y-5x\right)+15
Use the distributive property to multiply 3 by 3x-2y.
-6x-6y=5\left(3y-5x\right)+15
Combine 9x and -15x to get -6x.
-6x-6y=15y-25x+15
Use the distributive property to multiply 5 by 3y-5x.
-6x-6y-15y=-25x+15
Subtract 15y from both sides.
-6x-21y=-25x+15
Combine -6y and -15y to get -21y.
-6x-21y+25x=15
Add 25x to both sides.
19x-21y=15
Combine -6x and 25x to get 19x.
2\left(2x-3y\right)+30=3\left(4x-3y\right)+6y
Consider the second equation. Multiply both sides of the equation by 6, the least common multiple of 3,2.
4x-6y+30=3\left(4x-3y\right)+6y
Use the distributive property to multiply 2 by 2x-3y.
4x-6y+30=12x-9y+6y
Use the distributive property to multiply 3 by 4x-3y.
4x-6y+30=12x-3y
Combine -9y and 6y to get -3y.
4x-6y+30-12x=-3y
Subtract 12x from both sides.
-8x-6y+30=-3y
Combine 4x and -12x to get -8x.
-8x-6y+30+3y=0
Add 3y to both sides.
-8x-3y+30=0
Combine -6y and 3y to get -3y.
-8x-3y=-30
Subtract 30 from both sides. Anything subtracted from zero gives its negation.
19x-21y=15,-8x-3y=-30
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.
19x-21y=15
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
19x=21y+15
Add 21y to both sides of the equation.
x=\frac{1}{19}\left(21y+15\right)
Divide both sides by 19.
x=\frac{21}{19}y+\frac{15}{19}
Multiply \frac{1}{19} times 21y+15.
-8\left(\frac{21}{19}y+\frac{15}{19}\right)-3y=-30
Substitute \frac{21y+15}{19} for x in the other equation, -8x-3y=-30.
-\frac{168}{19}y-\frac{120}{19}-3y=-30
Multiply -8 times \frac{21y+15}{19}.
-\frac{225}{19}y-\frac{120}{19}=-30
Add -\frac{168y}{19} to -3y.
-\frac{225}{19}y=-\frac{450}{19}
Add \frac{120}{19} to both sides of the equation.
y=2
Divide both sides of the equation by -\frac{225}{19}, which is the same as multiplying both sides by the reciprocal of the fraction.
x=\frac{21}{19}\times 2+\frac{15}{19}
Substitute 2 for y in x=\frac{21}{19}y+\frac{15}{19}. Because the resulting equation contains only one variable, you can solve for x directly.
x=\frac{42+15}{19}
Multiply \frac{21}{19} times 2.
x=3
Add \frac{15}{19} to \frac{42}{19} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=3,y=2
The system is now solved.
3\left(3x-2y\right)-15x=5\left(3y-5x\right)+15
Consider the first equation. Multiply both sides of the equation by 15, the least common multiple of 5,3.
9x-6y-15x=5\left(3y-5x\right)+15
Use the distributive property to multiply 3 by 3x-2y.
-6x-6y=5\left(3y-5x\right)+15
Combine 9x and -15x to get -6x.
-6x-6y=15y-25x+15
Use the distributive property to multiply 5 by 3y-5x.
-6x-6y-15y=-25x+15
Subtract 15y from both sides.
-6x-21y=-25x+15
Combine -6y and -15y to get -21y.
-6x-21y+25x=15
Add 25x to both sides.
19x-21y=15
Combine -6x and 25x to get 19x.
2\left(2x-3y\right)+30=3\left(4x-3y\right)+6y
Consider the second equation. Multiply both sides of the equation by 6, the least common multiple of 3,2.
4x-6y+30=3\left(4x-3y\right)+6y
Use the distributive property to multiply 2 by 2x-3y.
4x-6y+30=12x-9y+6y
Use the distributive property to multiply 3 by 4x-3y.
4x-6y+30=12x-3y
Combine -9y and 6y to get -3y.
4x-6y+30-12x=-3y
Subtract 12x from both sides.
-8x-6y+30=-3y
Combine 4x and -12x to get -8x.
-8x-6y+30+3y=0
Add 3y to both sides.
-8x-3y+30=0
Combine -6y and 3y to get -3y.
-8x-3y=-30
Subtract 30 from both sides. Anything subtracted from zero gives its negation.
19x-21y=15,-8x-3y=-30
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}19&-21\\-8&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\-30\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}19&-21\\-8&-3\end{matrix}\right))\left(\begin{matrix}19&-21\\-8&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}19&-21\\-8&-3\end{matrix}\right))\left(\begin{matrix}15\\-30\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}19&-21\\-8&-3\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}19&-21\\-8&-3\end{matrix}\right))\left(\begin{matrix}15\\-30\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}19&-21\\-8&-3\end{matrix}\right))\left(\begin{matrix}15\\-30\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{3}{19\left(-3\right)-\left(-21\left(-8\right)\right)}&-\frac{-21}{19\left(-3\right)-\left(-21\left(-8\right)\right)}\\-\frac{-8}{19\left(-3\right)-\left(-21\left(-8\right)\right)}&\frac{19}{19\left(-3\right)-\left(-21\left(-8\right)\right)}\end{matrix}\right)\left(\begin{matrix}15\\-30\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{1}{75}&-\frac{7}{75}\\-\frac{8}{225}&-\frac{19}{225}\end{matrix}\right)\left(\begin{matrix}15\\-30\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{75}\times 15-\frac{7}{75}\left(-30\right)\\-\frac{8}{225}\times 15-\frac{19}{225}\left(-30\right)\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\2\end{matrix}\right)
Do the arithmetic.
x=3,y=2
Extract the matrix elements x and y.
3\left(3x-2y\right)-15x=5\left(3y-5x\right)+15
Consider the first equation. Multiply both sides of the equation by 15, the least common multiple of 5,3.
9x-6y-15x=5\left(3y-5x\right)+15
Use the distributive property to multiply 3 by 3x-2y.
-6x-6y=5\left(3y-5x\right)+15
Combine 9x and -15x to get -6x.
-6x-6y=15y-25x+15
Use the distributive property to multiply 5 by 3y-5x.
-6x-6y-15y=-25x+15
Subtract 15y from both sides.
-6x-21y=-25x+15
Combine -6y and -15y to get -21y.
-6x-21y+25x=15
Add 25x to both sides.
19x-21y=15
Combine -6x and 25x to get 19x.
2\left(2x-3y\right)+30=3\left(4x-3y\right)+6y
Consider the second equation. Multiply both sides of the equation by 6, the least common multiple of 3,2.
4x-6y+30=3\left(4x-3y\right)+6y
Use the distributive property to multiply 2 by 2x-3y.
4x-6y+30=12x-9y+6y
Use the distributive property to multiply 3 by 4x-3y.
4x-6y+30=12x-3y
Combine -9y and 6y to get -3y.
4x-6y+30-12x=-3y
Subtract 12x from both sides.
-8x-6y+30=-3y
Combine 4x and -12x to get -8x.
-8x-6y+30+3y=0
Add 3y to both sides.
-8x-3y+30=0
Combine -6y and 3y to get -3y.
-8x-3y=-30
Subtract 30 from both sides. Anything subtracted from zero gives its negation.
19x-21y=15,-8x-3y=-30
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.
-8\times 19x-8\left(-21\right)y=-8\times 15,19\left(-8\right)x+19\left(-3\right)y=19\left(-30\right)
To make 19x and -8x equal, multiply all terms on each side of the first equation by -8 and all terms on each side of the second by 19.
-152x+168y=-120,-152x-57y=-570
Simplify.
-152x+152x+168y+57y=-120+570
Subtract -152x-57y=-570 from -152x+168y=-120 by subtracting like terms on each side of the equal sign.
168y+57y=-120+570
Add -152x to 152x. Terms -152x and 152x cancel out, leaving an equation with only one variable that can be solved.
225y=-120+570
Add 168y to 57y.
225y=450
Add -120 to 570.
y=2
Divide both sides by 225.
-8x-3\times 2=-30
Substitute 2 for y in -8x-3y=-30. Because the resulting equation contains only one variable, you can solve for x directly.
-8x-6=-30
Multiply -3 times 2.
-8x=-24
Add 6 to both sides of the equation.
x=3
Divide both sides by -8.
x=3,y=2
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}