Question

Sketch the following sets of points in the \(x-y\) plane. $$ \left\\{(x, y) \in \mathbb{R}^{2}:(y-x)(y+x)=0\right\\} $$

Short answer

The solution is the two perpendicular lines forming an 'X' shape: \(y = x\) and \(y = -x\).

Step-by-step solution

Simplify the equation

The equation is \((y-x)(y+x) = 0\). This equation can be set to zero and factored out to be \(y = x\) and \(y = -x\)

Plot the first line

The line \(y = x\) is a straight line that passes through the origin (0,0) and has a positive slope (angle of 45 degrees with the x-axis). Plot this line.

Plot the second line

The line \(y = -x\) is also a straight line passing through the origin with a negative slope (angle of 135 degrees with the x-axis). Plot this line.

Visualize the solution

The combination of the two lines would give you an X shape. The point of intersection of the two lines is the origin.