Chapter 4 · Geometry and Equations · Concept 2
4-9 Application of the Equation of a Straight Line
Two question shapes, and neither needs a new formula. One asks whether three points sit on one line; the other asks for a line through the place two others cross. Both are lesson 4-8 with one step in front.
Guiding question
What has to be worked out before you can use what you already know?
Learning outcome. Use the equation of a straight line in geometric situations.
Nothing new, and one form to notice
An equation of the form ax + by + c = 0 (with a ≠ 0 or b ≠ 0) is also a straight line — the same one as y = mx + k, with everything moved to one side.
Three points on one line
Build the line from the two points you know completely, then substitute the third one in. Its coordinates must satisfy the equation, and that gives you the constant.
Through a crossing point
Solve the two given lines simultaneously to find where they cross. Then you have two points, and it is lesson 4-8.
Which form to hand in.
Match the question. If it wrote its lines as 2x − 3y + 1 = 0, answer in that form — the book's own margin note says so. Rearranging is one line of work and it is worth the mark.
The grey lines are what you are given. The answer cannot be drawn until the blue dot is known — which is why the simultaneous equations come first, every time.
2x − 3y + 1 = 0 and x + 2y − 3 = 0
1. A(2, 3), B(4, −1) and C(a − 1, 2 − a) lie on the same straight line. Find a.
- 1
Use the two points you know completely. C has letters in it, so it cannot help yet.
A(2, 3) and B(4, −1)Always build the line from the points with numbers. Trying to use C now would give one equation with two unknowns.
Work with a partner. For question 2, one of you solves the pair by substitution and the other by elimination. You should get the same crossing point — and you will find out which method suits which pair of numbers.
1–3. Three points on one line.
4–5. Through a crossing point.
Robotic needle biopsy. A tumour sits exactly where two diagnostic scan paths cross, on a grid where 1 unit = 1 millimetre:
6x + 5y + 7 = 0
3x − 2y + 5 = 0
The robot arm is anchored at the origin and must drive the needle in a straight line from there to the tumour. Find the equation of that path.
Building the line from the point with letters in it.
Use the two points you know completely first. The third one is what you substitute INTO the finished equation.
Trying to answer before finding the crossing.
Solve the two given lines simultaneously first. Until you have that point you only have one point, and one point does not fix a line.
Answering in the wrong form.
If the question wrote its lines as ax + by + c = 0, hand yours in that way too. It is one line of rearranging.
Confusing the two meanings of a.
The book uses a for the gradient in the formula AND a for the unknown constant. Call the gradient m in your own working if it helps.
Distrusting an ugly crossing point.
(−16, 13) and (−13/9, 1/3) are both correct answers. Check them in both original equations rather than assuming they are wrong.
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