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x-y=1,x+y=3269
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=1
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
x=y+1
Add y to both sides of the equation.
y+1+y=3269
Substitute y+1 for x in the other equation, x+y=3269.
2y+1=3269
Add y to y.
2y=3268
Subtract 1 from both sides of the equation.
y=1634
Divide both sides by 2.
x=1634+1
Substitute 1634 for y in x=y+1. Because the resulting equation contains only one variable, you can solve for x directly.
x=1635
Add 1 to 1634.
x=1635,y=1634
The system is now solved.
x-y=1,x+y=3269
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}1&-1\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\3269\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}1&-1\\1&1\end{matrix}\right))\left(\begin{matrix}1&-1\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&1\end{matrix}\right))\left(\begin{matrix}1\\3269\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-1\\1&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&-1\\1&1\end{matrix}\right))\left(\begin{matrix}1\\3269\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\\1&1\end{matrix}\right))\left(\begin{matrix}1\\3269\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-\left(-1\right)}&-\frac{-1}{1-\left(-1\right)}\\-\frac{1}{1-\left(-1\right)}&\frac{1}{1-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}1\\3269\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}{2}&\frac{1}{2}\\-\frac{1}{2}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}1\\3269\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}+\frac{1}{2}\times 3269\\-\frac{1}{2}+\frac{1}{2}\times 3269\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1635\\1634\end{matrix}\right)
Do the arithmetic.
x=1635,y=1634
Extract the matrix elements x and y.
x-y=1,x+y=3269
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.
x-x-y-y=1-3269
Subtract x+y=3269 from x-y=1 by subtracting like terms on each side of the equal sign.
-y-y=1-3269
Add x to -x. Terms x and -x cancel out, leaving an equation with only one variable that can be solved.
-2y=1-3269
Add -y to -y.
-2y=-3268
Add 1 to -3269.
y=1634
Divide both sides by -2.
x+1634=3269
Substitute 1634 for y in x+y=3269. Because the resulting equation contains only one variable, you can solve for x directly.
x=1635
Subtract 1634 from both sides of the equation.
x=1635,y=1634
The system is now solved.