How do I plot the graph of $|x|+|y|=1?$

Question

When I plot the graph of $|x|+|y|=1$ in Desmos Graphing Calculator it comes like this -

How do I graph this plot? Can someone guide me?

Answer 1

Since $|x|+|y|=1$, you have $|x|\leq 1$. Now separate in 4 cases :

• $x,y\geq 0$ : then $|x|+|y|=1$ can be written as $$y=1-x$$ for $0\leq x\leq 1$.
• $x\geq 0,y\leq 0$ : then $|x|+|y|=1$ can be written as $$y=x-1$$ for $0\leq x\leq 1$.
• $x\leq 0,y\geq 0$ : then $|x|+|y|=1$ can be written as $$y=1+x$$ for $-1\leq x\leq 0$.
• $x,y\leq 0$ : then $|x|+|y|=1$ can be written as $$y=-1-x$$ for $-1\leq x\leq 0$.

Answer 2

Break the equation $|x|+|y|=1$ as follows. Case 1: When $x\geq 0$, $y\geq 0$, then $|x|=x$ and $|y|=y$. Above equation turns into $x+y=1$ to give line segment AB.(By intercept form) Case 2: When $x\leq 0$, $y\geq 0$, then $|x|=-x$ and $|y|=y$. Above equation turns into $-x+y=1$ to give line segment AD. Case 3: When $x\leq 0$, $y\leq 0$, then $|x|=-x$ and $|y|=-y$. Above equation turns into $-x-y=1$ to give line segment DC. Case 4: When $x\geq 0$, $y\leq 0$, then $|x|=x$ and $|y|=-y$. Above equation turns into $x-y=1$ to give line segment CB. See the figure below