Solve for x, y
x=3100
y=3200
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x+y=6300,0.1x+0.07y=534
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=6300
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
x=-y+6300
Subtract y from both sides of the equation.
0.1\left(-y+6300\right)+0.07y=534
Substitute -y+6300 for x in the other equation, 0.1x+0.07y=534.
-0.1y+630+0.07y=534
Multiply 0.1 times -y+6300.
-0.03y+630=534
Add -\frac{y}{10} to \frac{7y}{100}.
-0.03y=-96
Subtract 630 from both sides of the equation.
y=3200
Divide both sides of the equation by -0.03, which is the same as multiplying both sides by the reciprocal of the fraction.
x=-3200+6300
Substitute 3200 for y in x=-y+6300. Because the resulting equation contains only one variable, you can solve for x directly.
x=3100
Add 6300 to -3200.
x=3100,y=3200
The system is now solved.
x+y=6300,0.1x+0.07y=534
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}1&1\\0.1&0.07\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6300\\534\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}1&1\\0.1&0.07\end{matrix}\right))\left(\begin{matrix}1&1\\0.1&0.07\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\0.1&0.07\end{matrix}\right))\left(\begin{matrix}6300\\534\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\0.1&0.07\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\\0.1&0.07\end{matrix}\right))\left(\begin{matrix}6300\\534\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\\0.1&0.07\end{matrix}\right))\left(\begin{matrix}6300\\534\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{0.07}{0.07-0.1}&-\frac{1}{0.07-0.1}\\-\frac{0.1}{0.07-0.1}&\frac{1}{0.07-0.1}\end{matrix}\right)\left(\begin{matrix}6300\\534\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{7}{3}&\frac{100}{3}\\\frac{10}{3}&-\frac{100}{3}\end{matrix}\right)\left(\begin{matrix}6300\\534\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{3}\times 6300+\frac{100}{3}\times 534\\\frac{10}{3}\times 6300-\frac{100}{3}\times 534\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3100\\3200\end{matrix}\right)
Do the arithmetic.
x=3100,y=3200
Extract the matrix elements x and y.
x+y=6300,0.1x+0.07y=534
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.
0.1x+0.1y=0.1\times 6300,0.1x+0.07y=534
To make x and \frac{x}{10} equal, multiply all terms on each side of the first equation by 0.1 and all terms on each side of the second by 1.
0.1x+0.1y=630,0.1x+0.07y=534
Simplify.
0.1x-0.1x+0.1y-0.07y=630-534
Subtract 0.1x+0.07y=534 from 0.1x+0.1y=630 by subtracting like terms on each side of the equal sign.
0.1y-0.07y=630-534
Add \frac{x}{10} to -\frac{x}{10}. Terms \frac{x}{10} and -\frac{x}{10} cancel out, leaving an equation with only one variable that can be solved.
0.03y=630-534
Add \frac{y}{10} to -\frac{7y}{100}.
0.03y=96
Add 630 to -534.
y=3200
Divide both sides of the equation by 0.03, which is the same as multiplying both sides by the reciprocal of the fraction.
0.1x+0.07\times 3200=534
Substitute 3200 for y in 0.1x+0.07y=534. Because the resulting equation contains only one variable, you can solve for x directly.
0.1x+224=534
Multiply 0.07 times 3200.
0.1x=310
Subtract 224 from both sides of the equation.
x=3100
Multiply both sides by 10.
x=3100,y=3200
The system is now solved.
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Matrix
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Simultaneous equation
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Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
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Limits
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