Testing a Point Against a Line

3 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

An equation does more than draw a line. It is a test.

The line $ax + by + c = 0$ is the set of all points whose coordinates make that equation true. So to ask "is this point on this line?", you do not need geometry. Put the numbers in and see if the equation holds.

Rule

The point $P(x_0, y_0)$ lies on the line $ax + by + c = 0$ exactly when

a x_0 + b y_0 + c = 0

Substitute and evaluate. Zero means yes; anything else means no.

How it is used

One substitution answers three questions that look different.

Question	What to do
Is $P$ on the line?	substitute, check the result is $0$
Find the missing coordinate	substitute the coordinate you have, solve for the other
Find the unknown in the equation	substitute the point, solve for the parameter

The middle row is how you find more points on a line. Choose any $x$ you like, and the equation gives you the matching $y$ .

The bottom row is the most common exam question. You are told a line passes through a point, and asked to find a letter inside the equation.

Be careful about what is unknown.

If the point is unknown, you solve for $x$ or $y$ . If the line is unknown, you solve for the letter in the equation, such as $k$ . The substitution looks the same either way, so read the question carefully.

Worked example

Determine whether $P(2, -1)$ lies on the line $3x + 2y - 4 = 0$ . Substitute $x = 2$ and $y = -1$ into the left-hand side of the equation:

3(2) + 2(-1) - 4 = 6 - 2 - 4 = 0

The left-hand side really does come out as zero, so the point satisfies the equation. Therefore $P(2, -1)$ lies on the line.

Explore

Put A and B anywhere and read the equation. Now choose any grid point that the blue line passes through and test it in that equation by hand. You will always get zero. Then try a point just off the line. The answer misses zero by exactly how far off you are.

A line through two points

Try this. Move the two points and compare the equation. Equal x values produce a vertical line; equal points cannot define one.

Point A · x-2Point A · y3Point B · x3Point B · y1

y = -0.4x + 2.2 · Slope = -0.4

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

(2,5)

on the line

y=3x-1

Only if the slope changes

Yes

Hint 1 · Find a starting point

A point is on a line exactly when its coordinates satisfy the equation.

Hint 2 · Take the next step

Evaluate 3(2) − 1 and compare with the point’s y-coordinate.

Show the reasoning

Answer: Yes

Yes: 3(2) − 1 = 5. This equality is an exact test, whereas a sketch can only suggest membership.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

Optional marks, not a grade. Saved in this browser only. Open notebook →