Home >> Algebra - Linear Graphs

**Coursework Notes - Algebra**

**Linear Graphs**

__The equation of a straight line__

This is given the form **y=mx+c**, where 'm' is the **gradient** of the graph and 'c' is the **intercept** on the y-axis(i.e. when x=0).

The **gradient(m)**of a line is the ratio of the 'y-step' to the 'x-step' from a consideration of two points on the line.

The **intercept - c** is the value of y when x=0. The other intercept(the value of x when y=0) is not used directly.

Since the equation of a straight line is **y=mx+c**, just looking at the equation is enough to give the gradient and the intercept on the y-axis.

**m** is the number infront of the x.

**c** is the number after the x term.

__Example #1__ Complete the table:

equation |
gradient |
intercept on y-axis |

y = x - 3 |
+1 |
-3 |

y = -3x + 4 |
-3 |
+4 |

y = 0.5x - 5 |
+0.5 |
-5 |

y + x = 1 |
-1 |
+1 |

x - y = 2 |
+1 |
-2 |

__Example #2__ Write down the equation of the straight line that goes through the points (2,1) and (5,7).

Putting into the equation one set of xy values(2,1),

hence the equation is:

__Parallel & perpendicular lines__

All lines with the same gradient are parallel. However, remember in each case the intercept with the x and y axis will be different.

Examples of parallel lines - note the value of 'c' in each case.

When two straight lines intersect at 90 degrees to eachother(i.e. are perpendicular), the product of their gradients is **-1**

__Example__ Complete the table of gradients of lines perpendicular to eachother.

line #1 gradient |
line #2 gradient |

1 |
-1 |

-2 |
0.5 |

3 |
-0.333 |

-4 |
0.25 |

5 |
-0.2 |

__The length of a line__ This is calculated using Pythagoras' Theorem.

The line between the two points is the hypotenuse of a right angled triangle.

Draw horizontal and vertical lines from the points.

Work out the lengths of the adjacent sides as you would to calculate gradient.

Then use Pythagoras to calculate the hypotenuse.

__Example__ Find the length of the line joining the points (1,2) and (3,5).

__The mid-point of a line__ - This is simply the average of the x-coordinate and the average of the y-coordinate.

__Example__ - if we take the two points from the last example, (1,2) (3,5), the mid point is:

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