Solve for x, y
x=400
y=100
Graph
Share
Copied to clipboard
x+y=500,50x+80y=28000
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=500
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
x=-y+500
Subtract y from both sides of the equation.
50\left(-y+500\right)+80y=28000
Substitute -y+500 for x in the other equation, 50x+80y=28000.
-50y+25000+80y=28000
Multiply 50 times -y+500.
30y+25000=28000
Add -50y to 80y.
30y=3000
Subtract 25000 from both sides of the equation.
y=100
Divide both sides by 30.
x=-100+500
Substitute 100 for y in x=-y+500. Because the resulting equation contains only one variable, you can solve for x directly.
x=400
Add 500 to -100.
x=400,y=100
The system is now solved.
x+y=500,50x+80y=28000
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}1&1\\50&80\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}500\\28000\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}1&1\\50&80\end{matrix}\right))\left(\begin{matrix}1&1\\50&80\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\50&80\end{matrix}\right))\left(\begin{matrix}500\\28000\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\50&80\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\\50&80\end{matrix}\right))\left(\begin{matrix}500\\28000\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\\50&80\end{matrix}\right))\left(\begin{matrix}500\\28000\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{80}{80-50}&-\frac{1}{80-50}\\-\frac{50}{80-50}&\frac{1}{80-50}\end{matrix}\right)\left(\begin{matrix}500\\28000\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{8}{3}&-\frac{1}{30}\\-\frac{5}{3}&\frac{1}{30}\end{matrix}\right)\left(\begin{matrix}500\\28000\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{3}\times 500-\frac{1}{30}\times 28000\\-\frac{5}{3}\times 500+\frac{1}{30}\times 28000\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}400\\100\end{matrix}\right)
Do the arithmetic.
x=400,y=100
Extract the matrix elements x and y.
x+y=500,50x+80y=28000
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.
50x+50y=50\times 500,50x+80y=28000
To make x and 50x equal, multiply all terms on each side of the first equation by 50 and all terms on each side of the second by 1.
50x+50y=25000,50x+80y=28000
Simplify.
50x-50x+50y-80y=25000-28000
Subtract 50x+80y=28000 from 50x+50y=25000 by subtracting like terms on each side of the equal sign.
50y-80y=25000-28000
Add 50x to -50x. Terms 50x and -50x cancel out, leaving an equation with only one variable that can be solved.
-30y=25000-28000
Add 50y to -80y.
-30y=-3000
Add 25000 to -28000.
y=100
Divide both sides by -30.
50x+80\times 100=28000
Substitute 100 for y in 50x+80y=28000. Because the resulting equation contains only one variable, you can solve for x directly.
50x+8000=28000
Multiply 80 times 100.
50x=20000
Subtract 8000 from both sides of the equation.
x=400
Divide both sides by 50.
x=400,y=100
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}