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Graphs of linear equations · 3 marks

Let us plot the points (− 3, 6), (− 2, 4), (0, 0), (1, − 2), (2, − 4), (3, − 6) in the coordinate plane on a graph paper as shown in the figure. Join the points (− 3, 6) and (3, − 6) using a ruler. Doing so, observe that all five points lie on a straight line. Can you guess the equation of this line by looking at the relationship between the x and y coordinates of each point?

xy−5−4−3−2−11234−6−4−22460(−3, 6)(−2, 4)(0, 0)(1, −2)(2, −4)(3, −6)
Answer: In every point the y-coordinate is −2 times the x-coordinate, so the line is y = −2x.

Step-by-step solution

Idea: Look for one rule that turns every x into its y. Here the signs are opposite and the size of y is double the size of x.

xy−5−4−3−2−11234−6−4−22460y = −2x(−3, 6)(−2, 4)(0, 0)(1, −2)(2, −4)(3, −6)
  1. Plot the points and join (−3, 6) to (3, −6) with a ruler: all the points lie on this straight line. (The question says “five points”, but six points are listed; all six lie on the line.)1 mark
  2. Compare: 6 = −2 × (−3), 4 = −2 × (−2), 0 = −2 × 0, −2 = −2 × 1, −4 = −2 × 2, −6 = −2 × 3.1 mark
  3. For each point, y is −2 times x. So the equation of the line is y = −2x. The line goes down from left to right.1 mark
The equation of the line is y = −2x.

Check: (−1, 2) is also on the drawn line, and 2 = −2 × (−1) ✓.

Answer to write in the exam

(−3, 6): 6 = −2(−3); (−2, 4): 4 = −2(−2); (0, 0): 0 = −2(0)

(1, −2): −2 = −2(1); (2, −4): −4 = −2(2); (3, −6): −6 = −2(3)

y-coordinate = −2 × x-coordinate for every point

∴ Equation of the line: y = −2x

Common mistakes that cost marks

  • Writing y = 2x, ignoring that x and y always have opposite signs.
  • Writing y = x − 2 from the point (1, −2) alone; it fails for (2, −4).
  • Plotting (1, −2) above the x-axis. A negative y-coordinate means below the x-axis.

How this can come in the exam

MCQ (1 mark)

The points (−1, 3), (0, 0), (2, −6) lie on the line

  1. y = 3x
  2. y = −3x
  3. y = x − 3
  4. y = −x + 3
Show answer

(B) y = −3x
3 = −3(−1), 0 = −3(0), −6 = −3(2).

Assertion–Reason (1 mark)

Assertion (A): The line y = −2x passes through the origin.
Reason (R): Every line of the form y = ax has y = 0 when x = 0.

  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
Show answer

(A) Both A and R are true, and R is the correct explanation of A.
Putting x = 0 gives y = 0 for any a, so (0, 0) lies on y = −2x. R explains A.

Try one yourself

Guess the equation of the line through (−4, 2), (0, 0), (2, −1) and (6, −3).

Show answer

Each y is −½ times x: y = −12x.

More questions like this

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