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. - 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

Our next step on our linear equation journey is to be able to interpret and solve problems involving equations of straight lines.

An equation of the form

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

has many names, depending on the state, country or even text book you use.

This equation is called:

- gradient-intercept formula
- slope-intercept formula

The values of $m$`m` and $b$`b` mean specific things. Remind 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.

So from equations in this form, $y=mx+b$`y`=`m``x`+`b`, we instantly have enough information to understand what this line looks like and to describe the transformations from the basic line $y=x$`y`=`x`.

By first identifying the gradient and $y$`y` intercept, describe the transformations of the following lines from the basic line $y=x$`y`=`x`.

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

- gradient is $3$3
- $y$
`y`intercept is $0$0 - Transformations of change: The line $y=x$
`y`=`x`is made steeper due to a gradient of $3$3 and is not vertically translated (it has the same $y$`y`-intercept as $y=x$`y`=`x`).

$y=-2x$`y`=−2`x`

- gradient is $-2$−2
- $y$
`y`intercept is $0$0 - Transformations of change: The line $y=x$
`y`=`x`is made steeper due to a gradient of $2$2, is reflected on the $x$`x`-axis (due to a negative gradient), and is not vertically translated.

$y=\frac{x}{2}-3$`y`=`x`2−3

- gradient is $\frac{1}{2}$12
- $y$
`y`intercept is $-3$−3 - Transformations of change: the line $y=x$
`y`=`x`is made less steep due to a gradient of $\frac{1}{2}$12 and is vertically translated $3$3 units down (a $y$`y`-intercept of $-3$−3 compared to a $y$`y`-intercept of $0$0 in $y=x$`y`=`x`).

$2y=-4x+10$2`y`=−4`x`+10

First we need to rewrite it in the gradient intercept form.

$y=-2x+5$`y`=−2`x`+5

- gradient is $-2$−2
- $y$
`y`intercept is $5$5 - Transformations of change: the line $y=x$
`y`=`x`is made more steep due to a gradient of $2$2, and is reflected on the $x$`x`-axis (due to a negative gradient). It is vertically translated $5$5 units up.

To create an equation of the form $y=mx+b$`y`=`m``x`+`b`, we need 2 pieces of information: if we know the gradient and the $y$`y`-intercept, we can instantly write down the equation.

What is the equations of the line with the a gradient of $\frac{3}{4}$34 and a $y$`y` intercept of $-2$−2?

The equation of the line will be:

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

$y=\frac{3}{4}x-2$`y`=34`x`−2

**It is easier to read the gradient and $y$ y-intercept from a linear equation if you rearrange the equation into gradient-intercept form:**

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

**What is the gradient of the line $y=\frac{3-2x}{8}$**`y`=3−2`x`8?

**Given that the line $y=mx+c$ y=mx+c has a gradient of $-2$−2 and passes through $\left(-6,-3\right)$(−6,−3):**

**Find $c$**`c`, the value of the $y$`y`-intercept of the line.**Find the equation of the line in the form $y=mx+c$**`y`=`m``x`+`c`.