\left\{ \begin{array} { l } { x + y = 13800 } \\ { 0.2 y = x } \end{array} \right.
Solve for x, y
x=2300
y=11500
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0.2y-x=0
Consider the second equation. Subtract x from both sides.
x+y=13800,-x+0.2y=0
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.
x+y=13800
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
x=-y+13800
Subtract y from both sides of the equation.
-\left(-y+13800\right)+0.2y=0
Substitute -y+13800 for x in the other equation, -x+0.2y=0.
y-13800+0.2y=0
Multiply -1 times -y+13800.
1.2y-13800=0
Add y to \frac{y}{5}.
1.2y=13800
Add 13800 to both sides of the equation.
y=11500
Divide both sides of the equation by 1.2, which is the same as multiplying both sides by the reciprocal of the fraction.
x=-11500+13800
Substitute 11500 for y in x=-y+13800. Because the resulting equation contains only one variable, you can solve for x directly.
x=2300
Add 13800 to -11500.
x=2300,y=11500
The system is now solved.
0.2y-x=0
Consider the second equation. Subtract x from both sides.
x+y=13800,-x+0.2y=0
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}1&1\\-1&0.2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}13800\\0\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}1&1\\-1&0.2\end{matrix}\right))\left(\begin{matrix}1&1\\-1&0.2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-1&0.2\end{matrix}\right))\left(\begin{matrix}13800\\0\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\-1&0.2\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}1&1\\-1&0.2\end{matrix}\right))\left(\begin{matrix}13800\\0\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}1&1\\-1&0.2\end{matrix}\right))\left(\begin{matrix}13800\\0\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{0.2}{0.2-\left(-1\right)}&-\frac{1}{0.2-\left(-1\right)}\\-\frac{-1}{0.2-\left(-1\right)}&\frac{1}{0.2-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}13800\\0\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}{6}&-\frac{5}{6}\\\frac{5}{6}&\frac{5}{6}\end{matrix}\right)\left(\begin{matrix}13800\\0\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\times 13800\\\frac{5}{6}\times 13800\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2300\\11500\end{matrix}\right)
Do the arithmetic.
x=2300,y=11500
Extract the matrix elements x and y.
0.2y-x=0
Consider the second equation. Subtract x from both sides.
x+y=13800,-x+0.2y=0
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.
-x-y=-13800,-x+0.2y=0
To make x and -x equal, multiply all terms on each side of the first equation by -1 and all terms on each side of the second by 1.
-x+x-y-0.2y=-13800
Subtract -x+0.2y=0 from -x-y=-13800 by subtracting like terms on each side of the equal sign.
-y-0.2y=-13800
Add -x to x. Terms -x and x cancel out, leaving an equation with only one variable that can be solved.
-1.2y=-13800
Add -y to -\frac{y}{5}.
y=11500
Divide both sides of the equation by -1.2, which is the same as multiplying both sides by the reciprocal of the fraction.
-x+0.2\times 11500=0
Substitute 11500 for y in -x+0.2y=0. Because the resulting equation contains only one variable, you can solve for x directly.
-x+2300=0
Multiply 0.2 times 11500.
-x=-2300
Subtract 2300 from both sides of the equation.
x=2300
Divide both sides by -1.
x=2300,y=11500
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
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Arithmetic
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Matrix
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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}