Find the equation of the line parallel to 2x + y = 7 that passes through (3, 5).
- Rearrange: y = −2x + 7, so the gradient is −2.
- The parallel line is y = −2x + c.
- Use (3, 5): 5 = −2 × 3 + c = −6 + c, so c = 11.
Answer: y = −2x + 11
Parallel lines
Maths · Coordinate geometry. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
Parallel lines go in the same direction and never meet. On a coordinate grid this means they have the same gradient. Their y-intercepts are different, otherwise they would be the same line.
To decide whether two lines are parallel, write each one as y = mx + c and compare the values of m. Do not compare the numbers in front of x until y is alone on one side.
To find the line parallel to a given line through a given point, copy the gradient of the given line, then use the point to find the new value of c.
Rules to remember
- Parallel lines have equal gradients: m1 = m2
- y = mx + c and y = mx + d are parallel when c and d are different
- Rearrange to y = mx + c before comparing gradients
- A line parallel to ax + by = c can be written ax + by = k; find k by substituting the point
- Line through (x1, y1) parallel to a line of gradient m: y − y1 = m(x − x1)
Common mistake
Taking the gradient straight from a form such as 2y = ax + 6. Its gradient is a/2, not a. Students also copy the old y-intercept; the new line needs its own c from the given point.
Worked example
Ask about this
Type what you are stuck on. The helper finds the explanation, worked examples and the notes to read, free and instant.
Read the full notes
Keep going
← Equations of linear graphsPerpendicular lines →Take a Maths test