Analytic Geometry

Ontario 09 Academic (MPM1D)

Equation of a line: Ax+By+C=0

Lesson

We have seen that the slope intercept form of a line looks like this:

$y=mx+b$`y`=`m``x`+`b`

From this form we can:

- instantly visualise where the line is positioned on the number plane
- identify if the line is increasing or decreasing
- see the value of the slope ($m$
`m`) - see the value of the $y$
`y`intercept ($b$`b`)

Another useful form for the equation of a straight line is the * general form*. It looks like this:

$ax+by+c=0$`a``x`+`b``y`+`c`=0

In this form:

- All coefficients $a$
`a`,$b$`b`and $c$`c`are integers - The coefficient of $x$
`x`is positive

Notice that in this form, the $y$`y`-intercept cannot be seen in the equation. We would have to substitute $x=0$`x`=0 to find it.

The advantages of writing an equation in this form can be seen when:

- there are fractions involved in the equation ($y=\frac{-5x}{3}-\frac{2}{7}$
`y`=−5`x`3−27 for example). Writing it in general form would be a tidier option. - we need to find the point of intersection of two straight lines (and one or both equations involve fractions)

We can convert from one form to another by rearranging the equation. Rearranging the equation is just like solving an equation: we carry out inverse operations to move terms from one side to another, or to change the sign from positive to negative. Let me show you what I mean.

Rearrange $y=4x-8$`y`=4`x`−8 into general form.

$y=4x-8$`y`=4`x`−8

Move all the terms to the same side, remembering to keep the coefficient of $x$`x` positive.

$4x-8-y=0$4`x`−8−`y`=0

Rearrange $3x-6y+12=0$3`x`−6`y`+12=0 into slope-intercept form.

$3x-6y+12=0$3`x`−6`y`+12=0

To make $y$`y` the subject, we need to move the $x$`x` term and the constant to the other side. It would be preferable to keep the coefficient of $y$`y` positive.

$3x+12=6y$3`x`+12=6`y`

We can now divide through by $6$6 to make $y$`y` the subject.

$\frac{1}{2}x+2$12x+2 |
$=$= | $y$y |

$y$y |
$=$= | $\frac{1}{2}x+2$12x+2 |

Which line is steeper, $2x+3y-2=0$2`x`+3`y`−2=0 or $2x+5y+3=0$2`x`+5`y`+3=0?

$2x+5y+3$2

`x`+5`y`+3$\text{ = }$ = $0$0A$\text{Both lines are equally steep. }$Both lines are equally steep. $\text{ }$ $\text{ }$

B$2x+3y-2$2

`x`+3`y`−2$\text{ = }$ = $0$0C$2x+5y+3$2

`x`+5`y`+3$\text{ = }$ = $0$0A$\text{Both lines are equally steep. }$Both lines are equally steep. $\text{ }$ $\text{ }$

B$2x+3y-2$2

`x`+3`y`−2$\text{ = }$ = $0$0C

Express the equation of a line in the form y = mx + b, given the form Ax + By + C = 0

Determine the equation of a line from information about the line