Sketch the set {(x,y) | x+y=3, x,y ∈ R}

Key concept: This is another linear relation in two variables. The fastest way to sketch it is to rewrite it in a standard form and use intercepts.

Step by step

Step 1: Rewrite the equation

Starting with

$x+y=3$

solve for $y$:

$y=3-x$

Step 2: Read off the line features

This is a straight line with:

• slope $-1$
• $y$-intercept $3$

Step 3: Find intercepts

Use the axes to sketch it accurately:

• If $x=0$, then $y=3$ → $(0,3)$
• If $y=0$, then $x=3$ → $(3,0)$

Step 4: Draw the line

Plot $(0,3)$ and $(3,0)$ and connect them with a straight line.

Final answer

The set is the line

$y=3-x$

passing through $(0,3)$ and $(3,0)$.

Pitfall alert

Do not plot only one point and guess the curve. Also, avoid writing the intercepts as $(3,0)$ and $(0,3)$ in the wrong order when interpreting the line's direction; the graph is determined by both points together.

Try different conditions

If the equation became $x+y=c$, the graph would still be a straight line with slope $-1$, but the intercepts would change to $(0,c)$ and $(c,0)$.

Further reading

intercepts, linear equation, slope