Parallel lines
Section: Coordinate geometry | Syllabus: Cambridge IGCSE Mathematics (0580)
Identifying and Finding Parallel Lines
Two straight lines are parallel if and only if they have the same gradient. This fact can be used to find the equation of a new line parallel to a given one.
- Parallel lines never meet, and always have exactly the same gradient
- To find the equation of a line parallel to a given line, first identify the gradient of the given line
- Then substitute this same gradient, along with the coordinates of the required point, into y = mx + c to find the new c
Worked Example: Finding the Equation of a Parallel Line
- Question: Find the equation of the line parallel to y = 3x - 4 that passes through the point (2, 9).
- Step 1: The given line has gradient 3, so the parallel line also has gradient 3: y=3x+c
- Step 2: Substitute the point (2, 9): 9=3(2)+c
- Step 3: Solve: 9=6+c c=3
- Answer: y = 3x + 3
Worked Example: Checking Whether Two Lines Are Parallel
- Question: Determine whether the lines 2y = 6x + 4 and y = 3x - 5 are parallel.
- Step 1: Rearrange the first line: 2y=6x+4 y=3x+2, gradient = 3
- Step 2: The second line, y = 3x - 5, also has gradient 3
- Answer: yes, the lines are parallel, since both have gradient 3
Common Mistakes
MistakeAssuming two lines with the same y-intercept must be parallel, rather than checking their gradients
Fixparallel lines are defined by having the same gradient - the y-intercept has no bearing on whether lines are parallel
MistakeComparing gradients before rearranging both equations into the form y = mx + c
Fixalways rearrange each equation fully before comparing coefficients - a line's gradient may be hidden until y is isolated
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