Practice Maths

y-intercept and Equation of a Line

Key Ideas

Key terms: y-intercept, slope-intercept form, gradient, equation of a line, sketching

• The y-intercept is the point where a line crosses the y-axis (where x = 0).
• The slope-intercept form of a linear equation is y = mx + b, where:
   — m is the gradient (slope)
   — b is the y-intercept
• To sketch a line from y = mx + b: plot the y-intercept (0, b), then use the gradient to find a second point.
• To find the equation from a graph: read off the y-intercept and calculate the gradient from two clear points.
• Note: In Queensland, you may also see the form y = mx + c, where c is the y-intercept. Both mean the same thing.
Hot Tip To sketch a line quickly from y = mx + b: start at the y-intercept (0, b). Then move right by the denominator of m and up (or down) by the numerator. Mark that second point and draw your line through both points.

Worked Example

Question A: Identify m and b in y = −3x + 7, then describe the line.

Answer: m = −3 (negative gradient, line slopes downward); b = 7 (y-intercept at (0, 7)).

Question B: Find the equation of the line with gradient 2 that passes through (0, −4).

Step 1 — The y-intercept is b = −4 (given that the line passes through (0, −4)).

Step 2 — Substitute m = 2 and b = −4 into y = mx + b.

Step 3 — Equation: y = 2x − 4

Question C: Find the equation of a line where the gradient is ½ and passes through (4, 3).

Step 1 — Substitute into y = mx + b: 3 = ½(4) + b → 3 = 2 + b → b = 1.

Step 2 — Equation: y = ½x + 1

The y-intercept: Where the Line Crosses the y-axis

The y-intercept is the point where a line crosses the vertical y-axis. At this point, x = 0. It represents the starting value of the relationship — what y equals before x has had any effect.

In the equation y = mx + b, the letter b is the y-intercept.

Real example: A mobile phone plan has a monthly fee of $25 and charges $0.15 per text message. Rule: Cost = 0.15 × texts + 25. The y-intercept is 25 — the cost before any texts are sent.

The Equation y = mx + b

This is the standard form for a linear equation:

• m = gradient (steepness, how fast y changes)
• b = y-intercept (starting value, where the line crosses the y-axis)

Examples of identifying m and b:
• y = 3x + 2 → m = 3, b = 2
• y = −x + 5 → m = −1, b = 5
• y = 4x → m = 4, b = 0 (line through the origin)
• y = 7 → m = 0, b = 7 (horizontal line)

Sketching a Line from Its Equation

To sketch y = mx + b without a full table of values, use two points:

Step 1: Plot the y-intercept: (0, b). This is always your first point.
Step 2: Use the gradient to find the next point. m = rise/run. From (0, b), go right by "run" and up by "rise" (or down if m is negative).
Step 3: Draw the line through both points with a ruler.

Example: Sketch y = 2x − 3.
b = −3: plot (0, −3).
m = 2 = 2/1: go right 1, up 2 → plot (1, −1).
Draw the line through (0, −3) and (1, −1).

Parallel Lines: Same Gradient, Different y-intercept

Two lines are parallel if they have the same gradient (m value) but different y-intercepts (b values). They never cross because they are always the same distance apart.

Example: y = 3x + 1 and y = 3x − 4 are parallel. Both have gradient 3, but one starts at y = 1 and the other at y = −4.

A line with y = 3x + 1 and another line y = 2x + 1 are NOT parallel — they have the same y-intercept but different gradients, so they cross at (0, 1).

Finding the Equation from a Graph

If you are given a graph and need to write the equation:

Step 1: Read off the y-intercept (where the line meets the y-axis) → this is b.
Step 2: Pick two clear points on the line and calculate gradient → this is m.
Step 3: Write y = mx + b.

Example: A line crosses the y-axis at (0, 4) and passes through (3, −2).
b = 4. m = (−2 − 4) ÷ (3 − 0) = −6 ÷ 3 = −2. Equation: y = −2x + 4.

Key tip: When sketching a line from y = mx + b, always start from the y-intercept (it's the easiest point to plot — just find it on the y-axis). Then use the gradient as directions: m = 3/1 means "right 1, up 3". m = −2/1 means "right 1, down 2". This makes sketching fast and accurate.

Mastery Practice

  1. Identify the gradient (m) and y-intercept (b) for each equation. Fluency

    1. y = 3x + 2
    2. y = −4x + 7
    3. y = 5x − 3
    4. y = −x + 9
    5. y = ½x − 6
    6. y = 2x
    7. y = −3
    8. y = −2x − 8
    9. y = ⅔x + 4
    10. y = −¾x + 1

  2. Write the equation y = mx + b given the gradient and y-intercept. Fluency

    1. m = 4, y-intercept = 1
    2. m = −2, y-intercept = 5
    3. m = 3, y-intercept = 0
    4. m = −1, y-intercept = −4
    5. m = ½, y-intercept = 3
    6. m = 0, y-intercept = −7
    7. m = −5, y-intercept = 2
    8. m = ⅔, y-intercept = −1

  3. Find the equation of the line (y = mx + b) given the gradient and a point on the line. Fluency

    1. m = 2, passes through (0, 5)
    2. m = −3, passes through (0, 1)
    3. m = 4, passes through (1, 6)
    4. m = −2, passes through (3, 4)
    5. m = 1, passes through (5, 3)
    6. m = ½, passes through (2, 0)
    7. m = 3, passes through (−1, 2)
    8. m = −4, passes through (2, −3)

  4. For each pair of lines, compare them by gradient and y-intercept. State whether they are parallel, the same, or different. Understanding

    1. y = 2x + 3 and y = 2x − 5
    2. y = 4x + 1 and y = −4x + 1
    3. y = −x + 6 and y = −x + 6
    4. y = 3x − 2 and y = 2x − 2
    5. y = ½x + 4 and y = 2x + 4
    6. y = 5x + 7 and y = 5x + 7

  5. For each graph description, write the equation of the line in y = mx + b form. Understanding

    1. A line crossing the y-axis at 4 and the x-axis at 2.
    2. A line crossing the y-axis at −3 and passing through (2, 1).
    3. A horizontal line passing through (0, −5).
    4. A line with the same y-intercept as y = 3x + 7 but gradient −2.
    5. A line through (0, 0) with gradient 5.
    6. A line through (3, 9) and (5, 13).

  6. Apply the equation of a line to real-world situations. Problem Solving

    1. A plumber charges a call-out fee plus an hourly rate. The total cost C (in dollars) is modelled by C = 80h + 60, where h is hours worked.
      1. What is the call-out fee?
      2. What is the hourly rate?
      3. How much does a 3-hour job cost?
    2. A candle is 24 cm tall when lit and burns down at 2 cm per hour. Write an equation for the height H of the candle after t hours. After how many hours is the candle 10 cm tall?
    3. Two lines are given: Line A has equation y = 3x − 6, and Line B passes through (1, −2) with gradient 3. Are Lines A and B the same line? Show your working.
    4. A linear function has the rule y = mx + b. It passes through (2, 7) and (5, 16). Find the equation of the line, then find the value of y when x = 10.
  7. Two streaming plans. Plan A costs $8 per month plus a $5 sign-up fee: CA = 8m + 5. Plan B has no sign-up fee and costs $11 per month: CB = 11m, where m is the number of months.
    1. What does the y-intercept represent for each plan?
    2. After how many months does Plan A become cheaper than Plan B? Show your working algebraically.
    3. What is the total cost of each plan after 12 months?
    Problem Solving

  8. Reading a line from a graph. Understanding

    The graph below shows the line y = 2x + 1. The y-intercept is marked with a dot.

    -5 -4 -3 -2 -1 1 2 3 4 5 5 4 3 2 1 -1 -2 -3 -4 -5 x y y = 2x + 1
    1. Write down the y-intercept from the graph.
    2. Choose two clear points on the line and use them to calculate the gradient.
    3. Write the equation of the line in y = mx + b form.
    4. Does the point (−2, −3) lie on this line? Check algebraically by substituting x = −2.

  9. Two lines on the same graph. Problem Solving

    The graph below shows two lines. Use the graph to answer the questions.

    -5 -4 -3 -2 -1 1 2 3 4 5 5 4 3 2 1 -1 -2 -3 -4 -5 x y A B
    1. Write down the y-intercept of Line A (blue) and Line B (red).
    2. Write down the gradient of Line A and Line B. (Use two clear points on each line.)
    3. Write the equation of each line in y = mx + b form.
    4. Estimate the coordinates of the intersection point from the graph. Then verify algebraically by setting the two equations equal.

  10. A line with a fractional gradient. Problem Solving

    The graph below shows a line. Use the graph to answer the questions.

    -5 -4 -3 -2 -1 1 2 3 4 5 5 4 3 2 1 -1 -2 -3 -4 -5 x y
    1. Identify the y-intercept from the graph.
    2. Read the rise and run between two clear grid points on the line to calculate the gradient. Show your working.
    3. Write the equation of the line in y = mx + b form.
    4. A student says “the gradient must be 2 because the line goes up slowly.” Identify the error in this reasoning and give the correct value.
    5. Find the x-intercept of the line (where y = 0) by solving your equation algebraically.
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