Solve for x, y
x=100
y=100
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x+4y=500,2x+y=300
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+4y=500
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
x=-4y+500
Subtract 4y from both sides of the equation.
2\left(-4y+500\right)+y=300
Substitute -4y+500 for x in the other equation, 2x+y=300.
-8y+1000+y=300
Multiply 2 times -4y+500.
-7y+1000=300
Add -8y to y.
-7y=-700
Subtract 1000 from both sides of the equation.
y=100
Divide both sides by -7.
x=-4\times 100+500
Substitute 100 for y in x=-4y+500. Because the resulting equation contains only one variable, you can solve for x directly.
x=-400+500
Multiply -4 times 100.
x=100
Add 500 to -400.
x=100,y=100
The system is now solved.
x+4y=500,2x+y=300
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}1&4\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}500\\300\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}1&4\\2&1\end{matrix}\right))\left(\begin{matrix}1&4\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\2&1\end{matrix}\right))\left(\begin{matrix}500\\300\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&4\\2&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}1&4\\2&1\end{matrix}\right))\left(\begin{matrix}500\\300\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&4\\2&1\end{matrix}\right))\left(\begin{matrix}500\\300\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}{1-4\times 2}&-\frac{4}{1-4\times 2}\\-\frac{2}{1-4\times 2}&\frac{1}{1-4\times 2}\end{matrix}\right)\left(\begin{matrix}500\\300\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}{7}&\frac{4}{7}\\\frac{2}{7}&-\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}500\\300\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{7}\times 500+\frac{4}{7}\times 300\\\frac{2}{7}\times 500-\frac{1}{7}\times 300\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}100\\100\end{matrix}\right)
Do the arithmetic.
x=100,y=100
Extract the matrix elements x and y.
x+4y=500,2x+y=300
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.
2x+2\times 4y=2\times 500,2x+y=300
To make x and 2x equal, multiply all terms on each side of the first equation by 2 and all terms on each side of the second by 1.
2x+8y=1000,2x+y=300
Simplify.
2x-2x+8y-y=1000-300
Subtract 2x+y=300 from 2x+8y=1000 by subtracting like terms on each side of the equal sign.
8y-y=1000-300
Add 2x to -2x. Terms 2x and -2x cancel out, leaving an equation with only one variable that can be solved.
7y=1000-300
Add 8y to -y.
7y=700
Add 1000 to -300.
y=100
Divide both sides by 7.
2x+100=300
Substitute 100 for y in 2x+y=300. Because the resulting equation contains only one variable, you can solve for x directly.
2x=200
Subtract 100 from both sides of the equation.
x=100
Divide both sides by 2.
x=100,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}