Does this help you to conclude anything about the linear equation y = ax + b when a is fixed but b varies?
(Hint: In these equations a = 2, and b takes the values −1, 1 and 5, respectively.)

Step-by-step solution

Idea: a controls the steepness and b controls where the line crosses the y-axis. Keep a the same and only the crossing point moves.

1. All three lines have a = 2, so they rise equally steeply and never meet: they are parallel.1 mark
2. They cut the y-axis at (0, −1), (0, 1) and (0, 5), that is, at (0, b). Increasing b shifts the line up; decreasing b shifts it down. So lines with equal slopes but different b (y-intercepts) are parallel.1 mark

Check: At any x, say x = 3, the three lines give 5, 7 and 11: the gaps (2 and 4) equal the differences in b, the same as at x = 0 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Saying the lines get steeper as b increases. Steepness is set by a, which is fixed.
• Saying b is where the line cuts the x-axis. It is where the line cuts the y-axis.
• Thinking parallel lines must pass through the origin.

How this can come in the exam

Try one yourself

Write the equation of the line parallel to y = 4x − 3 that passes through the origin.

More questions like this

• Now let us draw the graphs of the equations y = x + 3, y = 2x + 5 and y = 3x − 2. See the figure and observe where these lines cut the y-axis.
• Draw the graphs of the following sets of lines. In each case, reflect on the role of ‘a’ and ‘b’.
• Write a polynomial of degree 3 in the variable x, in which the coefficient of the x² term is −7.
• Find the values of the following polynomials at the indicated values of the variables.
• If we multiply a number by 52 and add 23 to the product, we get −712. Find the number.

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