Let us plot the points (−1, −3), (0, 0), (1, 3), (3, 9), (4, 12) in the coordinate plane on a graph paper as shown in the figure. Join the points (−1, −3) and (4, 12) 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?
Step-by-step solution
Idea: Compare y with x in each point. If the same rule turns every x into its y, that rule is the equation of the line.
- Plot the five points and join (−1, −3) to (4, 12) with a ruler: all five points lie on this straight line.1 mark
- Compare the coordinates: −3 = 3 × (−1), 0 = 3 × 0, 3 = 3 × 1, 9 = 3 × 3, 12 = 3 × 4.1 mark
- For each point, y is three times x. So the equation of the line is y = 3x.1 mark
Check: Another point on the line, (2, 6), also satisfies y = 3x, and the line passes through the origin as y = 3x must.
Answer to write in the exam
(−1, −3): −3 = 3(−1); (0, 0): 0 = 3(0); (1, 3): 3 = 3(1)
(3, 9): 9 = 3(3); (4, 12): 12 = 3(4)
y-coordinate = 3 × x-coordinate for every point
∴ Equation of the line: y = 3x
Common mistakes that cost marks
- Guessing y = x + 3 from the point (1, 3) alone. Test the rule on every point: (3, 9) gives 3 + 3 = 6 ≠ 9.
- Plotting (−1, −3) at (−3, −1) by swapping the coordinates.
- Writing x = 3y. It is y that is three times x.
How this can come in the exam
The points (−2, −8), (1, 4) and (3, 12) lie on the line
- y = 4x
- y = x + 3
- y = 2x + 2
- x = 4y
Show answer
(A) y = 4x
Each y is 4 times its x: −8 = 4(−2), 4 = 4(1), 12 = 4(3).
The points (0, 0), (2, 5), (4, 10) lie on a line through the origin. Find its equation and the y-coordinate of the point on it with x = 6.
Show answer
y = 52x (1 mark). At x = 6: y = 15 (1 mark).Try one yourself
Guess the equation of the line through (−2, −10), (0, 0), (1, 5) and (3, 15).
Show answer
Each y is 5 times x: y = 5x.
More questions like this
- 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?
- Draw the graphs of y = 12x, y = x, y = 2x by selecting suitable points on these lines.
(Hint: In order to graph y = 12x, we could take the points (0, 0) and (4, 2). Can you verify that these lie on the line?) - The figure shows all the three graphs on the same axes. Does this help you to conclude anything about the linear equation y = ax, a > 0 as a varies? What happens when a > 1 and when a < 1?
(Hint: You may also plot the equations y = 3x and y = 13x on the same axes.) - Now let us draw the graphs of y = −13x, y = −x, y = −3x by selecting suitable points on these lines.
- The figure shows all the three graphs on the same axes. Does this help you to conclude anything about the linear equation y = − ax, a > 0, as a varies? What will happen when a > 1 and when a < 1?