\left\{ \begin{array} { l } { x - 100 + \frac { x - y } { 2 } + 12 \cdot 5 = 0 } \\ { \frac { y - x } { 2 } - 125 + \frac { y } { 4 } + \frac { y + 50 } { 9 } = 0 } \end{array} \right.
Solve for x, y
x = \frac{452}{5} = 90\frac{2}{5} = 90.4
y = \frac{956}{5} = 191\frac{1}{5} = 191.2
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2x-200+x-y+24\times 5=0
Consider the first equation. Multiply both sides of the equation by 2.
3x-200-y+24\times 5=0
Combine 2x and x to get 3x.
3x-200-y+120=0
Multiply 24 and 5 to get 120.
3x-80-y=0
Add -200 and 120 to get -80.
3x-y=80
Add 80 to both sides. Anything plus zero gives itself.
18\left(y-x\right)-4500+9y+4\left(y+50\right)=0
Consider the second equation. Multiply both sides of the equation by 36, the least common multiple of 2,4,9.
18y-18x-4500+9y+4\left(y+50\right)=0
Use the distributive property to multiply 18 by y-x.
27y-18x-4500+4\left(y+50\right)=0
Combine 18y and 9y to get 27y.
27y-18x-4500+4y+200=0
Use the distributive property to multiply 4 by y+50.
31y-18x-4500+200=0
Combine 27y and 4y to get 31y.
31y-18x-4300=0
Add -4500 and 200 to get -4300.
31y-18x=4300
Add 4300 to both sides. Anything plus zero gives itself.
3x-y=80,-18x+31y=4300
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.
3x-y=80
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
3x=y+80
Add y to both sides of the equation.
x=\frac{1}{3}\left(y+80\right)
Divide both sides by 3.
x=\frac{1}{3}y+\frac{80}{3}
Multiply \frac{1}{3} times y+80.
-18\left(\frac{1}{3}y+\frac{80}{3}\right)+31y=4300
Substitute \frac{80+y}{3} for x in the other equation, -18x+31y=4300.
-6y-480+31y=4300
Multiply -18 times \frac{80+y}{3}.
25y-480=4300
Add -6y to 31y.
25y=4780
Add 480 to both sides of the equation.
y=\frac{956}{5}
Divide both sides by 25.
x=\frac{1}{3}\times \frac{956}{5}+\frac{80}{3}
Substitute \frac{956}{5} for y in x=\frac{1}{3}y+\frac{80}{3}. Because the resulting equation contains only one variable, you can solve for x directly.
x=\frac{956}{15}+\frac{80}{3}
Multiply \frac{1}{3} times \frac{956}{5} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
x=\frac{452}{5}
Add \frac{80}{3} to \frac{956}{15} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=\frac{452}{5},y=\frac{956}{5}
The system is now solved.
2x-200+x-y+24\times 5=0
Consider the first equation. Multiply both sides of the equation by 2.
3x-200-y+24\times 5=0
Combine 2x and x to get 3x.
3x-200-y+120=0
Multiply 24 and 5 to get 120.
3x-80-y=0
Add -200 and 120 to get -80.
3x-y=80
Add 80 to both sides. Anything plus zero gives itself.
18\left(y-x\right)-4500+9y+4\left(y+50\right)=0
Consider the second equation. Multiply both sides of the equation by 36, the least common multiple of 2,4,9.
18y-18x-4500+9y+4\left(y+50\right)=0
Use the distributive property to multiply 18 by y-x.
27y-18x-4500+4\left(y+50\right)=0
Combine 18y and 9y to get 27y.
27y-18x-4500+4y+200=0
Use the distributive property to multiply 4 by y+50.
31y-18x-4500+200=0
Combine 27y and 4y to get 31y.
31y-18x-4300=0
Add -4500 and 200 to get -4300.
31y-18x=4300
Add 4300 to both sides. Anything plus zero gives itself.
3x-y=80,-18x+31y=4300
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}3&-1\\-18&31\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}80\\4300\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}3&-1\\-18&31\end{matrix}\right))\left(\begin{matrix}3&-1\\-18&31\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\-18&31\end{matrix}\right))\left(\begin{matrix}80\\4300\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}3&-1\\-18&31\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}3&-1\\-18&31\end{matrix}\right))\left(\begin{matrix}80\\4300\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}3&-1\\-18&31\end{matrix}\right))\left(\begin{matrix}80\\4300\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{31}{3\times 31-\left(-\left(-18\right)\right)}&-\frac{-1}{3\times 31-\left(-\left(-18\right)\right)}\\-\frac{-18}{3\times 31-\left(-\left(-18\right)\right)}&\frac{3}{3\times 31-\left(-\left(-18\right)\right)}\end{matrix}\right)\left(\begin{matrix}80\\4300\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{31}{75}&\frac{1}{75}\\\frac{6}{25}&\frac{1}{25}\end{matrix}\right)\left(\begin{matrix}80\\4300\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{31}{75}\times 80+\frac{1}{75}\times 4300\\\frac{6}{25}\times 80+\frac{1}{25}\times 4300\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{452}{5}\\\frac{956}{5}\end{matrix}\right)
Do the arithmetic.
x=\frac{452}{5},y=\frac{956}{5}
Extract the matrix elements x and y.
2x-200+x-y+24\times 5=0
Consider the first equation. Multiply both sides of the equation by 2.
3x-200-y+24\times 5=0
Combine 2x and x to get 3x.
3x-200-y+120=0
Multiply 24 and 5 to get 120.
3x-80-y=0
Add -200 and 120 to get -80.
3x-y=80
Add 80 to both sides. Anything plus zero gives itself.
18\left(y-x\right)-4500+9y+4\left(y+50\right)=0
Consider the second equation. Multiply both sides of the equation by 36, the least common multiple of 2,4,9.
18y-18x-4500+9y+4\left(y+50\right)=0
Use the distributive property to multiply 18 by y-x.
27y-18x-4500+4\left(y+50\right)=0
Combine 18y and 9y to get 27y.
27y-18x-4500+4y+200=0
Use the distributive property to multiply 4 by y+50.
31y-18x-4500+200=0
Combine 27y and 4y to get 31y.
31y-18x-4300=0
Add -4500 and 200 to get -4300.
31y-18x=4300
Add 4300 to both sides. Anything plus zero gives itself.
3x-y=80,-18x+31y=4300
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.
-18\times 3x-18\left(-1\right)y=-18\times 80,3\left(-18\right)x+3\times 31y=3\times 4300
To make 3x and -18x equal, multiply all terms on each side of the first equation by -18 and all terms on each side of the second by 3.
-54x+18y=-1440,-54x+93y=12900
Simplify.
-54x+54x+18y-93y=-1440-12900
Subtract -54x+93y=12900 from -54x+18y=-1440 by subtracting like terms on each side of the equal sign.
18y-93y=-1440-12900
Add -54x to 54x. Terms -54x and 54x cancel out, leaving an equation with only one variable that can be solved.
-75y=-1440-12900
Add 18y to -93y.
-75y=-14340
Add -1440 to -12900.
y=\frac{956}{5}
Divide both sides by -75.
-18x+31\times \frac{956}{5}=4300
Substitute \frac{956}{5} for y in -18x+31y=4300. Because the resulting equation contains only one variable, you can solve for x directly.
-18x+\frac{29636}{5}=4300
Multiply 31 times \frac{956}{5}.
-18x=-\frac{8136}{5}
Subtract \frac{29636}{5} from both sides of the equation.
x=\frac{452}{5}
Divide both sides by -18.
x=\frac{452}{5},y=\frac{956}{5}
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}