Linear Equations

Lesson

Let's have a quick recap of what we know about straight lines on the Cartesian plane so far.

- They have a gradient (slope), a measure of how steep the line is.
- They can be increasing (positive gradient) or decreasing (negative gradient).
- They can be horizontal (zero gradient).
- They can be vertical (gradient is undefined).
- They have $x$
`x`intercepts, $y$`y`intercepts or both an $x$`x`and a $y$`y`intercept. - The gradient can be calculated using $\frac{\text{rise }}{\text{run }}$rise run or $\frac{y_2-y_1}{x_2-x_1}$
`y`2−`y`1`x`2−`x`1. - They have an equation of the form $y=mx+b$
`y`=`m``x`+`b`.

The values of $m$`m` and $b$`b` mean specific things. Explore for yourself what these values do by exploring on this interactive.

So what you will have found is that the $m$`m` value affects the gradient.

- If $m<0$
`m`<0, the gradient is negative and the line is decreasing - if $m>0$
`m`>0, the gradient is positive and the line is increasing - if $m=0$
`m`=0 the gradient is $0$0 and the line is horizontal - Also, the larger the value of $m$
`m`the steeper the line

We also found that the $b$`b` value affects the $y$`y` intercept.

- If $b$
`b`is positive then the line is vertically translated (moved) up. - If $b$
`b`is negative then the line is vertically translated (moved) down.

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

a) What is the the gradient? The gradient is the value of the coefficient, (the number in front of the $x$`x`).

The gradient of this line is 3.

b) What is the $y$`y`-intercept? The $y$`y`-intercept is the value of the constant term, (the number on its own). The $y$`y`-intercept of this line is 0.

$y=-2x$`y`=−2`x`

- gradient is $-2$−2
- $y$
`y`intercept is $0$0

$y=\frac{x}{2}-3$`y`=`x`2−3

- gradient is $\frac{1}{2}$12
- $y$
`y`intercept is $-3$−3

Consider the equation $y=-1-\frac{9x}{2}$`y`=−1−9`x`2.

State the gradient of the line.

State the $y$

`y`-value of the $y$`y`-intercept.

$2y=-4x+10$2`y`=−4`x`+10

First we need to rewrite it in the form or $y=mx+b$`y`=`m``x`+`b`.

$y=-2x+5$`y`=−2`x`+5

- gradient is $-2$−2
- $y$
`y`intercept is $5$5