Solve for x, y
x=39
y=35
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x+y=74,40x+60y=3660
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=74
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
x=-y+74
Subtract y from both sides of the equation.
40\left(-y+74\right)+60y=3660
Substitute -y+74 for x in the other equation, 40x+60y=3660.
-40y+2960+60y=3660
Multiply 40 times -y+74.
20y+2960=3660
Add -40y to 60y.
20y=700
Subtract 2960 from both sides of the equation.
y=35
Divide both sides by 20.
x=-35+74
Substitute 35 for y in x=-y+74. Because the resulting equation contains only one variable, you can solve for x directly.
x=39
Add 74 to -35.
x=39,y=35
The system is now solved.
x+y=74,40x+60y=3660
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}1&1\\40&60\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}74\\3660\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}1&1\\40&60\end{matrix}\right))\left(\begin{matrix}1&1\\40&60\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\40&60\end{matrix}\right))\left(\begin{matrix}74\\3660\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\40&60\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\\40&60\end{matrix}\right))\left(\begin{matrix}74\\3660\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\\40&60\end{matrix}\right))\left(\begin{matrix}74\\3660\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{60}{60-40}&-\frac{1}{60-40}\\-\frac{40}{60-40}&\frac{1}{60-40}\end{matrix}\right)\left(\begin{matrix}74\\3660\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&-\frac{1}{20}\\-2&\frac{1}{20}\end{matrix}\right)\left(\begin{matrix}74\\3660\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\times 74-\frac{1}{20}\times 3660\\-2\times 74+\frac{1}{20}\times 3660\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}39\\35\end{matrix}\right)
Do the arithmetic.
x=39,y=35
Extract the matrix elements x and y.
x+y=74,40x+60y=3660
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.
40x+40y=40\times 74,40x+60y=3660
To make x and 40x equal, multiply all terms on each side of the first equation by 40 and all terms on each side of the second by 1.
40x+40y=2960,40x+60y=3660
Simplify.
40x-40x+40y-60y=2960-3660
Subtract 40x+60y=3660 from 40x+40y=2960 by subtracting like terms on each side of the equal sign.
40y-60y=2960-3660
Add 40x to -40x. Terms 40x and -40x cancel out, leaving an equation with only one variable that can be solved.
-20y=2960-3660
Add 40y to -60y.
-20y=-700
Add 2960 to -3660.
y=35
Divide both sides by -20.
40x+60\times 35=3660
Substitute 35 for y in 40x+60y=3660. Because the resulting equation contains only one variable, you can solve for x directly.
40x+2100=3660
Multiply 60 times 35.
40x=1560
Subtract 2100 from both sides of the equation.
x=39
Divide both sides by 40.
x=39,y=35
The system is now solved.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
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}