Let's have a quick recap of what we have learnt about straight lines on the xy-plane so far.
They have a gradient (slope) which is 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-intercepts, y-intercepts or both an x and a y-intercept.
The gradient can be calculated using m=\dfrac{\text{rise}}{\text{run}} or m=\dfrac{y_2-y_1}{x_2-x_1}.
They have an equation of the form y=mx+c.
The values of m and c have specific meanings.
Explore for yourself what these values do by exploring on this interactive. Move the sliders and notice how m affects the gradient and c affects the y-intercept of the line.
We can see that the value of m affects the gradient, and that the value of c affects the y-intercept.
For the gradient:
If m<0, the gradient is negative and the line is decreasing.
If m>0, the gradient is positive and the line is increasing.
If m=0, the gradient is 0 and the line is horizontal.
The larger the magnitude of m the steeper the line.
For the y-intercept:
If c is positive then the line cross the y-axis above the origin.
If c is negative then the line cross the y-axis below the origin.
An equation of the form y=mx+c is known as the gradient-intercept form of a line, as we can easily identify both the gradient, m, and the value of the y-intercept, c.
Another useful form for the equation of a straight line is the general form. It looks like this:ax+by+c=0
In this form all coefficients a,b and c are integers and a is positive.
Notice that in this form, the y-intercept cannot be seen in the equation. We would have to substitute 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=\dfrac{-5x}{3}-\dfrac{2}{7} 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.
A line has the equation 3x-9y-27=0.
Express the equation of the line in the form y=mx+c.
What is the gradient of the line?
What is the y-intercept of the line?
The gradient-intercept form of a straight line:
The general form of a straight line:
Sometimes we are given the graph of a line, and we are asked to find the equation of the line.
The first thing we want to do is find the  gradient of the line , which we can do by using any two points (usually the intercepts). Using the coordinates of the two points, we can either use the gradient formula, or by calculating the rise and the run.
We can then identify the y-intercept, by looking where the line crosses the y-axis. Once we have identified these two features, we can write the equation of the line in the form y=mx+c.
Consider the line shown on the coordinate-plane:
Complete the table of values.
x | -1 | 0 | 1 | 2 |
---|---|---|---|---|
y |
Linear relations can be written in the form y=mx+c. For this relationship, state the values of m and c.
Write the linear equation expressing the relationship between x and y.
Find the coordinates for the point on the line where x=20.
Consider the line shown on the coordinate-plane:
State the value of the y-intercept.
By how much does the y-value change as the x-value increases by 1?
Write the linear equation expressing the relationship between x and y.
We can find the equation of a line of the form y=mx+c by finding the gradient for the value of m, and the point of y-intercept for the value of c.