Finding constants for a line or curve – Secrets of the Mathematical Ninja

The Mathematical Ninja stopped dead in his tracks. Not literally, of course: immortal beings don't die. He turned his head slowly towards the student, unsure what was going on.

He probably had in mind his recent OFSTED report, which had -- while praising the results of the students that survived for long enough to take exams -- made a few constructive suggestions about his teaching methods and their toll on the local A&E department.

More to the point, this was a new student; having checked his bank account, he'd discovered that the students he killed or maimed in the first lesson tended not to come back for a second. Instead, he decided to listen more carefully.

"I'm not sure how to find the constants on this curve," she said again, quite timidly.

The Mathematical Ninja took a deep breath. "What information do you have?"

"Well, I've got an equation for a curve - but it's got an $a$ and a $b$ in it, or else I'd sketch it!"

He nodded, and muttered: "sketching instinct, good."

"I also know that it goes through these two points."

"What can you do with those?"

The student thought better of shrugging, but managed to shrug with her eyebrows.

The Mathematical Ninja, too, internalised his stress relief and simply visualised a mediaeval torture device. That was better! "What do you call the numbers in the brackets, there?"

"Three and seven!" said the student, brightly.

He mentally turned a screw. "More generally?"

She thought for a moment. "Co-ordinates."

Tighter, still tighter. "Which co-ordinates?"

"Oh! $x$ and $y$?"

"And where else can you see an $x$ and a $y$?"

"In the other point?"

This was like pulling teeth, an image that pleased the Mathematical Ninja no end. "Alternatively? What are we trying to find?"

"Oh... in the equation!"

Finally. "Right! So...?"

"We could put the $x$ and the $y$ values into the equation... but we still have all those $a$s and $b$s!"

"Try it."

"OK... so I get two equations, each of them has an $a$ and a $b$ in. What now?"

The Mathematical Ninja tilted his head to one side. In most people, it would have looked cute.

"It can't be a simultaneous equation, it doesn't have an $x$ and a $y$ in..."

"I tried," said the Mathematical Ninja. "I tried SO HARD. Forgive me, OFSTED."


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.


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I teach in my home in Abbotsbury Road, Weymouth.

It's a 15-minute walk from Weymouth station, and it's on bus routes 3, 8 and X53. On-road parking is available nearby.

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