Linear Equations
UK Secondary (7-11)
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Gradient-Intercept Form
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 $\frac{y_2-y_1}{x_2-x_1}$y2y1x2x1

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

Gradient Intercept Form of a Straight Line

An equation of the form 

$y=mx+b$y=mx+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.

Gradient

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

Y-Intercept

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.

Transformations of the Line

So from equations in this form, $y=mx+b$y=mx+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.

Examples

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

Question 1

$y=3x$y=3x

  • 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).
 
Question 2

$y=-2x$y=2x

  • 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.
 
Question 3

$y=\frac{x}{2}-3$y=x23

  • 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).
 
Question 4

$2y=-4x+10$2y=4x+10

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

$y=-2x+5$y=2x+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.

 

Creating equations from information given about the line

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

Example

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=mx+b

$y=\frac{3}{4}x-2$y=34x2

Here are some worked examples.

Question 1

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=mx+b

  1. What is the gradient of the line $y=\frac{3-2x}{8}$y=32x8?

Question 2

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):

  1. Find $c$c, the value of the $y$y-intercept of the line.

  2. Find the equation of the line in the form $y=mx+c$y=mx+c.

 

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