Sketch the set {(x,y) | x=2y, x,y ∈ R}

Key concept: This set describes all points in the plane whose coordinates satisfy a simple linear relation. The key is to rewrite it in slope-intercept form and identify its geometric meaning.

Step by step

Step 1: Rewrite the equation

From

$x=2y$

solve for $y$:

$y=\frac{x}{2}$

Step 2: Identify the graph

This is a straight line with:

• slope $\frac12$
• $y$-intercept $0$

So the line passes through the origin.

Step 3: Plot a few points

Choose convenient values:

• if $y=0$, then $x=0$ → $(0,0)$
• if $y=1$, then $x=2$ → $(2,1)$
• if $y=-1$, then $x=-2$ → $(-2,-1)$

Step 4: Draw the sketch

Connect these points with a straight line extending in both directions.

Final answer

The set is the line

$y=\frac{x}{2}$

through the origin.

Pitfall alert

A common mistake is to treat $x=2y$ as a curved graph or to swap the slope and write $y=2x$. The correct slope is $\frac12$, not $2$.

Try different conditions

If the equation were changed to $x=2y+1$, the graph would still be a line, but it would shift and no longer pass through the origin. If it were restricted to $x,y\ge 0$, only the first-quadrant part of the line would remain.

Further reading

Cartesian plane, linear equation, slope-intercept form