Analytic Geometry

Lesson

Straight lines are lines on the Cartesian Plane that extend forever in both directions. If we ignore for a moment the special cases of horizontal and vertical lines, straight lines will cross **both **the $x$`x`-axis and the $y$`y`-axis or maybe through the point where the $x$`x` and $y$`y` axes cross (called the origin).

Here are some examples...

The word **intercept **in mathematics refers to a point where a line, curve or function crosses or intersects with the axes.

- We can have $x$
`x`intercepts: where the line, curve or function crosses the $x$`x`axis. - We can have $y$
`y`intercepts: where the line, curve or function crosses the $y$`y`axis.

Consider what happens as you move up or down along the $y$`y`-axis. You eventually reach the origin ($\left(0,0\right)$(0,0)) where $y=0$`y`=0. Now, if you move along the $x$`x`-axis in either direction, the $y$`y` value is still $0$0.

Similarly, consider what happens as you move along the $x$`x`-axis. You eventually reach the origin where $x=0$`x`=0. Now, if you move along the $y$`y`-axis in either direction, the $x$`x` value is still $0$0.

So, two important properties are:

- any point on the $x$
`x`-axis will have $y$`y`value of $0$0 - any point on the $y$
`y`-axis will have $x$`x`value of $0$0

We can use these properties to calculate or identify $x$`x` and $y$`y` intercepts for any line, curve or function.

Intercepts

The $x$`x` intercept occurs at the point where $y=0$`y`=0.

The $y$`y` intercept occurs at the point where $x=0$`x`=0.

Find the $x$`x` and $y$`y` intercepts for the following lines.

$y=3x$`y`=3`x`

**Think**: The $x$`x` intercept occurs when $y=0$`y`=0. The $y$`y` intercept occurs when $x=0$`x`=0.

**Do**: When $x=0$`x`=0, $y=3\times0$`y`=3×0 = $0$0

This means that this line passes through $\left(0,0\right)$(0,0), the origin. The $x$`x` and $y$`y` intercept occur at the same point!

*This particular form of a straight line *$y=mx$`y`=`m``x` *always passes through the origin. *

$y=4x-7$`y`=4`x`−7

**Think**: The $x$`x` intercept occurs when $y=0$`y`=0. The $y$`y` intercept occurs when $x=0$`x`=0.

**Do**: When $x=0$`x`=0

$y=4\times0-7$`y`=4×0−7 = $-7$−7 So the $y$`y` intercept is $-7$−7

When $y=0$`y`=0

$0$0 | $=$= | $4x-7$4x−7 |

$7$7 | $=$= | $4x$4x |

$\frac{7}{4}$74 | $=$= | $x$x |

So the $x$`x` intercept is $\frac{7}{4}$74

*This form of a straight line* $y=mx+b$`y`=`m``x`+`b`, *always has* $y$`y` *intercept of* $b$`b`.

*The $x$ x intercept is easy to work out after that (substitute* $y=0$

$2y-5x-10=0$2`y`−5`x`−10=0

**Think**: The $x$`x` intercept occurs when $y=0$`y`=0. The $y$`y` intercept occurs when $x=0$`x`=0.

**Do**: When $x=0$`x`=0, the $5x$5`x` term disappears. This leaves us with:

$2y-10$2y−10 |
$=$= | $0$0 |

$2y$2y |
$=$= | $10$10 |

$y$y |
$=$= | $5$5 |

So the $y$`y` intercept is $5$5

When $y=0$`y`=0, the $2y$2`y` term disappears. This leaves us with:

$-5x-10$−5x−10 |
$=$= | $0$0 |

$-5x$−5x |
$=$= | $10$10 |

$x$x |
$=$= | $-2$−2 |

So the $x$`x` intercept is $-2$−2

Let's have a look at some worked solutions.

**What is the $x$ x-value of the $x$x-intercept of the line $-3x+4y=-27$−3x+4y=−27?**

**What is the $y$ y-value of the $y$y-intercept of the line with equation: $-5x+3y=27$−5x+3y=27**

**Consider the points in the table:**

Time in minutes ($x$x) |
1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Temperature in °C ($y$y) |
$6$6 | $9$9 | $12$12 | $15$15 | $18$18 |

**By how much is the temperature increasing each minute?****What would the temperature have been at time 0?****Find the algebraic relationship between $x$**`x`and $y$`y`.

Determine, through investigation, various formulas for the slope of a line segment or to determine the slope of a line segment or a line