\left\{ \begin{array} { l } { x + y = 200 } \\ { 4.2 x + 3.4 y = 3.6 \times 200 } \end{array} \right.
Solve for x, y
x=50
y=150
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x+y=200,4.2x+3.4y=720
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=200
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
x=-y+200
Subtract y from both sides of the equation.
4.2\left(-y+200\right)+3.4y=720
Substitute -y+200 for x in the other equation, 4.2x+3.4y=720.
-4.2y+840+3.4y=720
Multiply 4.2 times -y+200.
-0.8y+840=720
Add -\frac{21y}{5} to \frac{17y}{5}.
-0.8y=-120
Subtract 840 from both sides of the equation.
y=150
Divide both sides of the equation by -0.8, which is the same as multiplying both sides by the reciprocal of the fraction.
x=-150+200
Substitute 150 for y in x=-y+200. Because the resulting equation contains only one variable, you can solve for x directly.
x=50
Add 200 to -150.
x=50,y=150
The system is now solved.
x+y=200,4.2x+3.4y=720
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}1&1\\4.2&3.4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}200\\720\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}1&1\\4.2&3.4\end{matrix}\right))\left(\begin{matrix}1&1\\4.2&3.4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\4.2&3.4\end{matrix}\right))\left(\begin{matrix}200\\720\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\4.2&3.4\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\\4.2&3.4\end{matrix}\right))\left(\begin{matrix}200\\720\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\\4.2&3.4\end{matrix}\right))\left(\begin{matrix}200\\720\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{3.4}{3.4-4.2}&-\frac{1}{3.4-4.2}\\-\frac{4.2}{3.4-4.2}&\frac{1}{3.4-4.2}\end{matrix}\right)\left(\begin{matrix}200\\720\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}-4.25&1.25\\5.25&-1.25\end{matrix}\right)\left(\begin{matrix}200\\720\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4.25\times 200+1.25\times 720\\5.25\times 200-1.25\times 720\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\150\end{matrix}\right)
Do the arithmetic.
x=50,y=150
Extract the matrix elements x and y.
x+y=200,4.2x+3.4y=720
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.
4.2x+4.2y=4.2\times 200,4.2x+3.4y=720
To make x and \frac{21x}{5} equal, multiply all terms on each side of the first equation by 4.2 and all terms on each side of the second by 1.
4.2x+4.2y=840,4.2x+3.4y=720
Simplify.
4.2x-4.2x+4.2y-3.4y=840-720
Subtract 4.2x+3.4y=720 from 4.2x+4.2y=840 by subtracting like terms on each side of the equal sign.
4.2y-3.4y=840-720
Add \frac{21x}{5} to -\frac{21x}{5}. Terms \frac{21x}{5} and -\frac{21x}{5} cancel out, leaving an equation with only one variable that can be solved.
0.8y=840-720
Add \frac{21y}{5} to -\frac{17y}{5}.
0.8y=120
Add 840 to -720.
y=150
Divide both sides of the equation by 0.8, which is the same as multiplying both sides by the reciprocal of the fraction.
4.2x+3.4\times 150=720
Substitute 150 for y in 4.2x+3.4y=720. Because the resulting equation contains only one variable, you can solve for x directly.
4.2x+510=720
Multiply 3.4 times 150.
4.2x=210
Subtract 510 from both sides of the equation.
x=50
Divide both sides of the equation by 4.2, which is the same as multiplying both sides by the reciprocal of the fraction.
x=50,y=150
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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