Solve for a, s
s=19
a=54
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2a+3s=165,3a+2s=200
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.
2a+3s=165
Choose one of the equations and solve it for a by isolating a on the left hand side of the equal sign.
2a=-3s+165
Subtract 3s from both sides of the equation.
a=\frac{1}{2}\left(-3s+165\right)
Divide both sides by 2.
a=-\frac{3}{2}s+\frac{165}{2}
Multiply \frac{1}{2} times -3s+165.
3\left(-\frac{3}{2}s+\frac{165}{2}\right)+2s=200
Substitute \frac{-3s+165}{2} for a in the other equation, 3a+2s=200.
-\frac{9}{2}s+\frac{495}{2}+2s=200
Multiply 3 times \frac{-3s+165}{2}.
-\frac{5}{2}s+\frac{495}{2}=200
Add -\frac{9s}{2} to 2s.
-\frac{5}{2}s=-\frac{95}{2}
Subtract \frac{495}{2} from both sides of the equation.
s=19
Divide both sides of the equation by -\frac{5}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.
a=-\frac{3}{2}\times 19+\frac{165}{2}
Substitute 19 for s in a=-\frac{3}{2}s+\frac{165}{2}. Because the resulting equation contains only one variable, you can solve for a directly.
a=\frac{-57+165}{2}
Multiply -\frac{3}{2} times 19.
a=54
Add \frac{165}{2} to -\frac{57}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
a=54,s=19
The system is now solved.
2a+3s=165,3a+2s=200
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}2&3\\3&2\end{matrix}\right)\left(\begin{matrix}a\\s\end{matrix}\right)=\left(\begin{matrix}165\\200\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}2&3\\3&2\end{matrix}\right))\left(\begin{matrix}2&3\\3&2\end{matrix}\right)\left(\begin{matrix}a\\s\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&2\end{matrix}\right))\left(\begin{matrix}165\\200\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&3\\3&2\end{matrix}\right).
\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\s\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&2\end{matrix}\right))\left(\begin{matrix}165\\200\end{matrix}\right)
The product of a matrix and its inverse is the identity matrix.
\left(\begin{matrix}a\\s\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&2\end{matrix}\right))\left(\begin{matrix}165\\200\end{matrix}\right)
Multiply the matrices on the left hand side of the equal sign.
\left(\begin{matrix}a\\s\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2\times 2-3\times 3}&-\frac{3}{2\times 2-3\times 3}\\-\frac{3}{2\times 2-3\times 3}&\frac{2}{2\times 2-3\times 3}\end{matrix}\right)\left(\begin{matrix}165\\200\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}a\\s\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{5}&\frac{3}{5}\\\frac{3}{5}&-\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}165\\200\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}a\\s\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{5}\times 165+\frac{3}{5}\times 200\\\frac{3}{5}\times 165-\frac{2}{5}\times 200\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}a\\s\end{matrix}\right)=\left(\begin{matrix}54\\19\end{matrix}\right)
Do the arithmetic.
a=54,s=19
Extract the matrix elements a and s.
2a+3s=165,3a+2s=200
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.
3\times 2a+3\times 3s=3\times 165,2\times 3a+2\times 2s=2\times 200
To make 2a and 3a equal, multiply all terms on each side of the first equation by 3 and all terms on each side of the second by 2.
6a+9s=495,6a+4s=400
Simplify.
6a-6a+9s-4s=495-400
Subtract 6a+4s=400 from 6a+9s=495 by subtracting like terms on each side of the equal sign.
9s-4s=495-400
Add 6a to -6a. Terms 6a and -6a cancel out, leaving an equation with only one variable that can be solved.
5s=495-400
Add 9s to -4s.
5s=95
Add 495 to -400.
s=19
Divide both sides by 5.
3a+2\times 19=200
Substitute 19 for s in 3a+2s=200. Because the resulting equation contains only one variable, you can solve for a directly.
3a+38=200
Multiply 2 times 19.
3a=162
Subtract 38 from both sides of the equation.
a=54
Divide both sides by 3.
a=54,s=19
The system is now solved.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
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}