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Lesson 13 of 14

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Ten ideas in the table, then nine questions. The table comes first, so nothing has to be recalled cold.

Ten ways two straight lines can answer a question, or refuse to
IdeaHow it goes
Two lettersax + by + c = 0. Both letters have power 1, never multiplied. A solution is a pair.
Pass twicePut the pair in both equations. One level row is not a verdict.
One lineEvery pair of one equation lies on one line, and every point of it.
Two linesEach equation draws a line. The crossing is the solution.
The read-offOn a grid mark, check it in both equations. Between marks it is an estimate.
Three verdictsCross once, run parallel, or lie on one line: one, none, all.
SubstitutionName one letter, use its name in the other equation, solve.
Scale, then subtractMatch one letter's coefficients, then subtract it away.
Both roadsSubstitution and elimination cannot land on two answers.
The leftoverA letter ends it. A false sentence refuses. 0 = 0 means one line.

Which of these is a solution of 2x + 3y = 12?

  • 4, because 3 × 4 = 12
  • the pair (4, 4)
  • the pair (1.5, 3)

A pair satisfies the first equation of a pair of equations. What is known about the second?

  • Nothing yet. Check the second
  • It is satisfied too, since the first one was
  • It must give the same pair

The lines of 2x + 3y = 12 and x − y = 1 are drawn on one plane. What does the crossing give?

  • Two solutions, one from each line
  • The one pair that fits both equations
  • Only a rough hint, even on a grid mark

A drawn crossing sits between grid marks. What is the honest reading?

  • The exact answer, read off the page
  • Proof that the lines never meet
  • Only an estimate. Use algebra

What do the lines of 2x + 3y = 12 and 2x + 3y = 18 do?

  • Cross once, at a negative pair
  • Run parallel and never meet, so no solution
  • Lie on top of each other

Substitution on 2x + 3y = 12 and x − y = 1 starts with x = y + 1. Why is that move allowed?

  • It keeps every solution of the equation
  • x agreed to stand in for y
  • It works only because the numbers are small

To remove y from 3x + 2y = 370 and x + y = 140, what comes first?

  • Subtract straight away, no scaling
  • Scale the second by 3, then subtract
  • Scale the second by 2, then subtract

Substitution and elimination are both run on one pair. What can happen?

  • Each road can print its own answer
  • They agree on one pair, none, or one line
  • Substitution always finishes first

Eliminating a letter ends at 0 = 0. What have you learned?

  • One line, said twice
  • The pair has no solution at all
  • There is exactly one solution, the pair (0, 0)

This topic is Linear Equations in Two Variables. Keep its ideas beside your Class 9 book or notes.

Next: Introduction to Euclid's Geometry: Axioms and Postulates. Every proof leans on it.

Reveal every answer to finish this lesson.