Two distinct parallel lines with defined gradients have the same gradient. Read the gradient from the given equation, then equate it to the rise divided by the run between the two points on the other line.
The 2027 SEC G3 Mathematics K310 syllabus covers calculating a line’s gradient from two points and interpreting straight-line equations. These examples show how to use both skills when one coordinate is missing.
- Read the coefficient of x in the form y = mx + c.
- Use the same endpoint order for both coordinate differences.
- Equate the gradients and solve for the missing coordinate.
- Substitute your answer into the gradient calculation.
CHAPTER 1 OF 4 · Back to contents
Separate slope from intercept
In y = mx + c, m is the gradient and c is the y-intercept. For distinct parallel lines, the slopes match and the intercepts differ when both lines have defined gradients.
Between two points, gradient is the change in y divided by the change in x. Use destination minus start for both changes.
A negative gradient means the line falls as x increases. Predict the direction of the missing coordinate before calculating.
CHAPTER 2 OF 4 · Back to contents
Worked example | Find k for a line through (3, 5) and (7, k)
The line through these points is parallel to y = −2x + 7. Its gradient must equal −2. From the first point to the second, x increases by 4.
| Quantity | Working | Result |
|---|---|---|
| Given gradient | Coefficient of x | −2 |
| Horizontal change | 7 − 3 | 4 |
| Vertical change | −2 × 4 | −8 |
| Missing y-coordinate | 5 − 8 | −3 |
Equivalently, write (k − 5)/4 = −2. Multiply by 4 to get k − 5 = −8, so k = −3.
Check that (−3 − 5)/(7 − 3) = −2. The point (3, 5) is off the original line because that line has y = 1 at x = 3. The two lines are distinct.
Contents · Previous chapter · Next chapter
CHAPTER 3 OF 4 · Back to contents
Practise with a fractional gradient
A line through (−2, 1) and (6, t) is parallel to y = ½x + 5. The run is 8, so the rise is ½ × 8 = 4. Hence t = 1 + 4 = 5.
Check the sign using the sketch. The second point is to the right and should be higher because the gradient is positive.
If the given line is vertical, use its constant x-coordinate as the clue. Any distinct parallel vertical line has another constant x-coordinate.
Contents · Previous chapter · Next chapter
CHAPTER 4 OF 4 · Back to contents
Frequently asked questions
What if the given equation is in another form?
Rearrange it into y = mx + c when the gradient is defined, then read m.
Can I reverse the point order?
Yes. Reverse both differences so the signs cancel in the ratio.
What if both lines have the same gradient and intercept?
They describe the same line.
Contents · Previous chapter · Continue with the related guide