Solve for x, y
x=400
y=600
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x+y=1000,8x+1.7y=4220
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=1000
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
x=-y+1000
Subtract y from both sides of the equation.
8\left(-y+1000\right)+1.7y=4220
Substitute -y+1000 for x in the other equation, 8x+1.7y=4220.
-8y+8000+1.7y=4220
Multiply 8 times -y+1000.
-6.3y+8000=4220
Add -8y to \frac{17y}{10}.
-6.3y=-3780
Subtract 8000 from both sides of the equation.
y=600
Divide both sides of the equation by -6.3, which is the same as multiplying both sides by the reciprocal of the fraction.
x=-600+1000
Substitute 600 for y in x=-y+1000. Because the resulting equation contains only one variable, you can solve for x directly.
x=400
Add 1000 to -600.
x=400,y=600
The system is now solved.
x+y=1000,8x+1.7y=4220
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}1&1\\8&1.7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1000\\4220\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}1&1\\8&1.7\end{matrix}\right))\left(\begin{matrix}1&1\\8&1.7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\8&1.7\end{matrix}\right))\left(\begin{matrix}1000\\4220\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\8&1.7\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\\8&1.7\end{matrix}\right))\left(\begin{matrix}1000\\4220\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\\8&1.7\end{matrix}\right))\left(\begin{matrix}1000\\4220\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{1.7}{1.7-8}&-\frac{1}{1.7-8}\\-\frac{8}{1.7-8}&\frac{1}{1.7-8}\end{matrix}\right)\left(\begin{matrix}1000\\4220\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{17}{63}&\frac{10}{63}\\\frac{80}{63}&-\frac{10}{63}\end{matrix}\right)\left(\begin{matrix}1000\\4220\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{17}{63}\times 1000+\frac{10}{63}\times 4220\\\frac{80}{63}\times 1000-\frac{10}{63}\times 4220\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}400\\600\end{matrix}\right)
Do the arithmetic.
x=400,y=600
Extract the matrix elements x and y.
x+y=1000,8x+1.7y=4220
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.
8x+8y=8\times 1000,8x+1.7y=4220
To make x 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 1.
8x+8y=8000,8x+1.7y=4220
Simplify.
8x-8x+8y-1.7y=8000-4220
Subtract 8x+1.7y=4220 from 8x+8y=8000 by subtracting like terms on each side of the equal sign.
8y-1.7y=8000-4220
Add 8x to -8x. Terms 8x and -8x cancel out, leaving an equation with only one variable that can be solved.
6.3y=8000-4220
Add 8y to -\frac{17y}{10}.
6.3y=3780
Add 8000 to -4220.
y=600
Divide both sides of the equation by 6.3, which is the same as multiplying both sides by the reciprocal of the fraction.
8x+1.7\times 600=4220
Substitute 600 for y in 8x+1.7y=4220. Because the resulting equation contains only one variable, you can solve for x directly.
8x+1020=4220
Multiply 1.7 times 600.
8x=3200
Subtract 1020 from both sides of the equation.
x=400
Divide both sides by 8.
x=400,y=600
The system is now solved.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
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Linear equation
y = 3x + 4
Arithmetic
699 * 533
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}