Solve for x, y
x=-105000
y=110000
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x+y=5000,0.05x+0.075y=3000
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=5000
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
x=-y+5000
Subtract y from both sides of the equation.
0.05\left(-y+5000\right)+0.075y=3000
Substitute -y+5000 for x in the other equation, 0.05x+0.075y=3000.
-0.05y+250+0.075y=3000
Multiply 0.05 times -y+5000.
0.025y+250=3000
Add -\frac{y}{20} to \frac{3y}{40}.
0.025y=2750
Subtract 250 from both sides of the equation.
y=110000
Multiply both sides by 40.
x=-110000+5000
Substitute 110000 for y in x=-y+5000. Because the resulting equation contains only one variable, you can solve for x directly.
x=-105000
Add 5000 to -110000.
x=-105000,y=110000
The system is now solved.
x+y=5000,0.05x+0.075y=3000
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}1&1\\0.05&0.075\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5000\\3000\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}1&1\\0.05&0.075\end{matrix}\right))\left(\begin{matrix}1&1\\0.05&0.075\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\0.05&0.075\end{matrix}\right))\left(\begin{matrix}5000\\3000\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\0.05&0.075\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\\0.05&0.075\end{matrix}\right))\left(\begin{matrix}5000\\3000\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\\0.05&0.075\end{matrix}\right))\left(\begin{matrix}5000\\3000\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{0.075}{0.075-0.05}&-\frac{1}{0.075-0.05}\\-\frac{0.05}{0.075-0.05}&\frac{1}{0.075-0.05}\end{matrix}\right)\left(\begin{matrix}5000\\3000\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}3&-40\\-2&40\end{matrix}\right)\left(\begin{matrix}5000\\3000\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\times 5000-40\times 3000\\-2\times 5000+40\times 3000\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-105000\\110000\end{matrix}\right)
Do the arithmetic.
x=-105000,y=110000
Extract the matrix elements x and y.
x+y=5000,0.05x+0.075y=3000
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.
0.05x+0.05y=0.05\times 5000,0.05x+0.075y=3000
To make x and \frac{x}{20} equal, multiply all terms on each side of the first equation by 0.05 and all terms on each side of the second by 1.
0.05x+0.05y=250,0.05x+0.075y=3000
Simplify.
0.05x-0.05x+0.05y-0.075y=250-3000
Subtract 0.05x+0.075y=3000 from 0.05x+0.05y=250 by subtracting like terms on each side of the equal sign.
0.05y-0.075y=250-3000
Add \frac{x}{20} to -\frac{x}{20}. Terms \frac{x}{20} and -\frac{x}{20} cancel out, leaving an equation with only one variable that can be solved.
-0.025y=250-3000
Add \frac{y}{20} to -\frac{3y}{40}.
-0.025y=-2750
Add 250 to -3000.
y=110000
Multiply both sides by -40.
0.05x+0.075\times 110000=3000
Substitute 110000 for y in 0.05x+0.075y=3000. Because the resulting equation contains only one variable, you can solve for x directly.
0.05x+8250=3000
Multiply 0.075 times 110000.
0.05x=-5250
Subtract 8250 from both sides of the equation.
x=-105000
Multiply both sides by 20.
x=-105000,y=110000
The system is now solved.
Examples
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Matrix
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Simultaneous equation
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Differentiation
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Integration
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Limits
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