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18x-3y=72,36x+3y=360
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.
18x-3y=72
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
18x=3y+72
Add 3y to both sides of the equation.
x=\frac{1}{18}\left(3y+72\right)
Divide both sides by 18.
x=\frac{1}{6}y+4
Multiply \frac{1}{18} times 72+3y.
36\left(\frac{1}{6}y+4\right)+3y=360
Substitute \frac{y}{6}+4 for x in the other equation, 36x+3y=360.
6y+144+3y=360
Multiply 36 times \frac{y}{6}+4.
9y+144=360
Add 6y to 3y.
9y=216
Subtract 144 from both sides of the equation.
y=24
Divide both sides by 9.
x=\frac{1}{6}\times 24+4
Substitute 24 for y in x=\frac{1}{6}y+4. Because the resulting equation contains only one variable, you can solve for x directly.
x=4+4
Multiply \frac{1}{6} times 24.
x=8
Add 4 to 4.
x=8,y=24
The system is now solved.
18x-3y=72,36x+3y=360
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}18&-3\\36&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}72\\360\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}18&-3\\36&3\end{matrix}\right))\left(\begin{matrix}18&-3\\36&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}18&-3\\36&3\end{matrix}\right))\left(\begin{matrix}72\\360\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}18&-3\\36&3\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}18&-3\\36&3\end{matrix}\right))\left(\begin{matrix}72\\360\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}18&-3\\36&3\end{matrix}\right))\left(\begin{matrix}72\\360\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}{18\times 3-\left(-3\times 36\right)}&-\frac{-3}{18\times 3-\left(-3\times 36\right)}\\-\frac{36}{18\times 3-\left(-3\times 36\right)}&\frac{18}{18\times 3-\left(-3\times 36\right)}\end{matrix}\right)\left(\begin{matrix}72\\360\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{1}{54}&\frac{1}{54}\\-\frac{2}{9}&\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}72\\360\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{54}\times 72+\frac{1}{54}\times 360\\-\frac{2}{9}\times 72+\frac{1}{9}\times 360\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\24\end{matrix}\right)
Do the arithmetic.
x=8,y=24
Extract the matrix elements x and y.
18x-3y=72,36x+3y=360
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.
36\times 18x+36\left(-3\right)y=36\times 72,18\times 36x+18\times 3y=18\times 360
To make 18x and 36x equal, multiply all terms on each side of the first equation by 36 and all terms on each side of the second by 18.
648x-108y=2592,648x+54y=6480
Simplify.
648x-648x-108y-54y=2592-6480
Subtract 648x+54y=6480 from 648x-108y=2592 by subtracting like terms on each side of the equal sign.
-108y-54y=2592-6480
Add 648x to -648x. Terms 648x and -648x cancel out, leaving an equation with only one variable that can be solved.
-162y=2592-6480
Add -108y to -54y.
-162y=-3888
Add 2592 to -6480.
y=24
Divide both sides by -162.
36x+3\times 24=360
Substitute 24 for y in 36x+3y=360. Because the resulting equation contains only one variable, you can solve for x directly.
36x+72=360
Multiply 3 times 24.
36x=288
Subtract 72 from both sides of the equation.
x=8
Divide both sides by 36.
x=8,y=24
The system is now solved.