Parallel line equations

Two lines are parallel if they will never meet, no matter how far they are extended.

The key thing to remember when looking at their equations is that:

parallel lines have the same gradient

From this knowledge, we can find the equation of any line if we know that it’s parallel to a line that we know, and we know a point that the new line goes through.

Example: find the equation of the line parallel to y = 2x + 3 that goes through the point (4, 5)

• Gradient of original line is 2

• So the gradient of the new line is also 2

• We know that one of the points on the new line is (4, 5), so when x=4, y=5.

• We can substitute these values into the equation of a straight line:

  • y = mx + c
  • 5 = 2 \times 4 + c
  • 5 = 8 + c
  • c = 5 - 8
  • c = -3

• The y-intercept (c) is -3, so write the equation for a line with gradient 2 and y-intercept -3:

  • y = 2x - 3

• Answer: y = 2x - 3

Example: find the equation of the line parallel to y = -\frac{1}{2}x + 4 that goes through the point (6, 1)

• Gradient of original line is -\frac{1}{2}

• So the gradient of the new line is also -\frac{1}{2}

• We know that one of the points on the new line is (6, 1), so when x=6, y=1.

• We can substitute these values into the equation of a straight line:

  • y = mx + c
  • 1 = -\frac{1}{2} \times 6 + c
  • 1 = -3 + c
  • c = 1 + 3
  • c = 4

• The y-intercept (c) is 4, so write the equation for a line with gradient -\frac{1}{2} and y-intercept 4:

  • y = -\frac{1}{2}x + 4

• Answer: y = -\frac{1}{2}x + 4

Example: find the equation of the line parallel to y = 3x - 2 that goes through the point (2, 7)

• Gradient of original line is 3

• So the gradient of the new line is also 3

• We know that one of the points on the new line is (2, 7), so when x=2, y=7.

• We can substitute these values into the equation of a straight line:

  • y = mx + c
  • 7 = 3 \times 2 + c
  • 7 = 6 + c
  • c = 7 - 6
  • c = 1

• The -intercept (c) is 1, so write the equation for a line with gradient 3 and y-intercept 1:

  • y = 3x + 1

• Answer: y = 3x + 1

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