In this section we consider the particular relationship between the equations of parallel lines and perpendicular lines. The key to this is the *gradient* of lines that are parallel or perpendicular to each other.

(a)

Draw the lines with equations

y = x | y = x + 4 | y = x – 2 |

(b)

What do the three equations have in common?

Note that the three lines are parallel, all with gradient 1. All the equations of the lines contain '*x*'. This is because the gradient of each line is 1, and so the value of *m* in the equation *y* = *mx* + *c* is always 1.

For example, the lines with equations:

y = | 4x – 2 |

y = | 4x |

y = | 4x + 10 |

The equations of four lines are listed below:

A | y = 3x + 2 | B | y = 4x + 2 |
---|---|---|---|

C | y = 3x – 8 | D | y = 4x + 12 |

(a)

Which line is parallel to A ?

C is parallel to A, because both equations contain 3*x* (the coefficient of *x* in both cases is 3).

(b)

Which line is parallel to B ?

D is parallel to B, because both equations contain 4*x* (the coefficient of *x* in both cases is 4).

The graph shows two perpendicular lines, A and B:

(a)

Calculate the gradient of A and write down its equation.

Intercept of A = –7

Equation of A is*y* = 2*x* – 7

Gradient of A | = | |

= | 2 |

Equation of A is

(b)

Calculate the gradient of B and write down its equation.

Intercept of B = –2

Gradient of B | = | |

= |

Equation of B is y = | x – 2 |

(c)

Describe how the gradients of the lines are related.

So:

The gradients of the lines are 2 and | . |

Gradient of B = |

If two lines A and B are *perpendicular*,

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OR | ||||

Gradient of A × Gradient of B = –1 |

Line A has equation *y* = 3*x* + 2. Write down the gradient of line B that is perpendicular, and a possible equation for B.

Gradient of A | = | 3 |

Gradient of B | = | |

= |

Equation of B will be y = | x + c. |

so a possible equation is y = | x + 4. |