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150x-\frac{4}{5}y=0
Consider the first equation. Subtract \frac{4}{5}y from both sides.
200x-200=y+25
Consider the second equation. Use the distributive property to multiply 200 by x-1.
200x-200-y=25
Subtract y from both sides.
200x-y=25+200
Add 200 to both sides.
200x-y=225
Add 25 and 200 to get 225.
150x-\frac{4}{5}y=0,200x-y=225
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.
150x-\frac{4}{5}y=0
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
150x=\frac{4}{5}y
Add \frac{4y}{5} to both sides of the equation.
x=\frac{1}{150}\times \frac{4}{5}y
Divide both sides by 150.
x=\frac{2}{375}y
Multiply \frac{1}{150} times \frac{4y}{5}.
200\times \frac{2}{375}y-y=225
Substitute \frac{2y}{375} for x in the other equation, 200x-y=225.
\frac{16}{15}y-y=225
Multiply 200 times \frac{2y}{375}.
\frac{1}{15}y=225
Add \frac{16y}{15} to -y.
y=3375
Multiply both sides by 15.
x=\frac{2}{375}\times 3375
Substitute 3375 for y in x=\frac{2}{375}y. Because the resulting equation contains only one variable, you can solve for x directly.
x=18
Multiply \frac{2}{375} times 3375.
x=18,y=3375
The system is now solved.
150x-\frac{4}{5}y=0
Consider the first equation. Subtract \frac{4}{5}y from both sides.
200x-200=y+25
Consider the second equation. Use the distributive property to multiply 200 by x-1.
200x-200-y=25
Subtract y from both sides.
200x-y=25+200
Add 200 to both sides.
200x-y=225
Add 25 and 200 to get 225.
150x-\frac{4}{5}y=0,200x-y=225
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}150&-\frac{4}{5}\\200&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\225\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}150&-\frac{4}{5}\\200&-1\end{matrix}\right))\left(\begin{matrix}150&-\frac{4}{5}\\200&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}150&-\frac{4}{5}\\200&-1\end{matrix}\right))\left(\begin{matrix}0\\225\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}150&-\frac{4}{5}\\200&-1\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}150&-\frac{4}{5}\\200&-1\end{matrix}\right))\left(\begin{matrix}0\\225\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}150&-\frac{4}{5}\\200&-1\end{matrix}\right))\left(\begin{matrix}0\\225\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}{150\left(-1\right)-\left(-\frac{4}{5}\times 200\right)}&-\frac{-\frac{4}{5}}{150\left(-1\right)-\left(-\frac{4}{5}\times 200\right)}\\-\frac{200}{150\left(-1\right)-\left(-\frac{4}{5}\times 200\right)}&\frac{150}{150\left(-1\right)-\left(-\frac{4}{5}\times 200\right)}\end{matrix}\right)\left(\begin{matrix}0\\225\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}{10}&\frac{2}{25}\\-20&15\end{matrix}\right)\left(\begin{matrix}0\\225\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{25}\times 225\\15\times 225\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}18\\3375\end{matrix}\right)
Do the arithmetic.
x=18,y=3375
Extract the matrix elements x and y.
150x-\frac{4}{5}y=0
Consider the first equation. Subtract \frac{4}{5}y from both sides.
200x-200=y+25
Consider the second equation. Use the distributive property to multiply 200 by x-1.
200x-200-y=25
Subtract y from both sides.
200x-y=25+200
Add 200 to both sides.
200x-y=225
Add 25 and 200 to get 225.
150x-\frac{4}{5}y=0,200x-y=225
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.
200\times 150x+200\left(-\frac{4}{5}\right)y=0,150\times 200x+150\left(-1\right)y=150\times 225
To make 150x and 200x equal, multiply all terms on each side of the first equation by 200 and all terms on each side of the second by 150.
30000x-160y=0,30000x-150y=33750
Simplify.
30000x-30000x-160y+150y=-33750
Subtract 30000x-150y=33750 from 30000x-160y=0 by subtracting like terms on each side of the equal sign.
-160y+150y=-33750
Add 30000x to -30000x. Terms 30000x and -30000x cancel out, leaving an equation with only one variable that can be solved.
-10y=-33750
Add -160y to 150y.
y=3375
Divide both sides by -10.
200x-3375=225
Substitute 3375 for y in 200x-y=225. Because the resulting equation contains only one variable, you can solve for x directly.
200x=3600
Add 3375 to both sides of the equation.
x=18
Divide both sides by 200.
x=18,y=3375
The system is now solved.