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6 Linear relationships
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Level 8

6.03 Graphs of linear equations

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Lesson

Introduction

We have looked at how to  visualise a linear relationship  on a number plane, and we learnt that we only actually need to identify two points on the number plane in order to sketch the line. We will now look at how to sketch a line directly from its equation, without needing to create a table of values first.

In  Linear rules  we learnt that all linear relationships can be expressed in the form: y=mx+c, where m is equal to the change in the y-values for every increase in the x-value by 1, and c is the value of y when x=0.

Intercepts

Lines drawn on the number plane, extend forever in both directions. If we ignore the special case of horizontal and vertical lines (which we will look at in another lesson), all other lines will either cross both the x-axis and the y-axis or they will pass through the origin, (0,0).

Here are some examples:

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A line with an x-intercept of -3 and y-intercept of 2
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A line with an x and y-intercept of 0

We use the word intercept to refer to the point where the line crosses or intercepts with an axis.

  • The x-intercept is the point where the line crosses the x-axis. The coordinates of the x-intercept will always have a y-coordinate of zero.

  • The y-intercept is the point where the line crosses the y-axis. The coordinates of the y-intercept will always have an x-coordinate of zero.

Note: Every straight line must have at least one intercept but cannot have any more than two intercepts.

As mentioned previously, we only need to identify two points to sketch a a straight line, and the x and y-intercepts are probably the most useful points to identify and plot. They are also two of the easier points to find as we are substituting in either the values x=0 or y=0, which simplifies the work needed to solve.

The y-intercept can be thought of as either the co-ordinate the point where the y-axis is crossed, or simply the y-value at this point (as the x-value is by default 0).

Examples

Example 1

Consider the following graph.

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a

State the x-value of the x-intercept.

Worked Solution
Create a strategy

The x-intercept is the point where the line intersects the x-axis.

Apply the idea

The line intersects the x-axis at (3,0). So the x-value of the x-intercept is x=3.

b

State the y-value of the y-intercept.

Worked Solution
Create a strategy

The y-intercept is the point where the line intersects the y-axis.

Apply the idea

The line intersects the y-axis at (0,4). So the y-value of the y-intercept is y=4.

Example 2

A line has the equation y=3x-3.

a

Find the y-value of the y-intercept.

Worked Solution
Create a strategy

Substitute x=0 into the given equation.

Apply the idea
\displaystyle y\displaystyle =\displaystyle 3\times(0)-3Substitute x=0
\displaystyle =\displaystyle -3Evaluate
b

Find the x-value of the x-intercept.

Worked Solution
Create a strategy

Substitute y=0 into the given equation.

Apply the idea
\displaystyle 0\displaystyle =\displaystyle 3x-3Substitute y=0
\displaystyle 3x-3+3\displaystyle =\displaystyle 0+3Add 3 to both sides
\displaystyle 3x\displaystyle =\displaystyle 3Evaluate
\displaystyle \dfrac {3x}{3}\displaystyle =\displaystyle \dfrac {3}{3}Divide both sides by 3
\displaystyle x\displaystyle =\displaystyle 1Evaluate
c

Sketch the line of the equation y=3x-3.

Worked Solution
Create a strategy

Plot the intercepts and draw a straight line through them.

Apply the idea
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We can see that the intercepts (1,0) and (0,-3) were plotted, and then a straight line was drawn through the points.

Idea summary

  • The x-intercept is the point where the line crosses the x-axis. The coordinates of the x-intercept will always have a y-coordinate of zero.

  • The y-intercept is the point where the line crosses the y-axis. The coordinates of the y-intercept will always have an x-coordinate of zero.

Note: Every straight line must have at least one intercept but cannot have any more than two intercepts.

The gradient of a line

The change in y-values for every increase in the x-value is called the gradient. The gradient is often thought of as the 'slope' of the line - how steep it is.

The value of the gradient, m, relates to the line as follows:

  • A negative gradient (m<0) means the line is decreasing.

  • A positive gradient (m>0) means the line is increasing.

  • A zero gradient (m=0) means the line is horizontal.

  • The higher the value of m, the steeper the line.

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An increasing line (m\gt 0)
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A decreasing line (m\lt 0)
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A horizontal line (m=0)

Examples

Example 3

What is the gradient m of the line y=9x+3?

Worked Solution
Create a strategy

For all linear equations of the form y=mx+c, m is the gradient of the line.

Apply the idea

m=9

Idea summary

The value of the gradient, m, relates to the line as follows:

  • A negative gradient (m<0) means the line is decreasing

  • A positive gradient (m>0) means the line is increasing

  • A zero gradient (m=0) means the line is horizontal

  • The higher the value of m, the steeper the line.

Gradient-intercept form of a straight line

Any straight line on the coordinate plane is defined entirely by its gradient and its y-intercept.

We can represent the equation of any straight line, except vertical lines, using what is known as the gradient-intercept form of a straight line.

All linear relationships can be expressed in the form: y=mx+c

  • m is equal to the gradient, or slope, of the line.

  • c is the value of the y-intercept.

Exploration

We can use the applet below to see the effect of varying m and c on both the line and its equation.

Loading interactive...

The value of c is the y-intercept while m shows how steep the line of the equation is.

As m increases, the steepness of the line increases. As c increases, the line is shifted upwards.

In algebra, any number written immediately in front of a variable, is called a coefficient. For example, in the term 3x, the coefficient of x is 3. Any number by itself is known as a constant term.

In the gradient-intercept form of a line, y=mx+c, the gradient, m, is the coefficient of x, and the y-intercept, c, is a constant term.

Examples

Example 4

Consider the linear equation y=2x+9.

What are the values of the gradient, m and the y-intercept, c?

Worked Solution
Create a strategy

m is the coefficient of x and c is the constant term.

Apply the idea

The coefficient of x is 2. So m=2.

The constant term is 9. So c=9.

Idea summary

All linear relationships can be expressed in the form:

\displaystyle y=mx+c
\bm{m}
is equal to the gradient, or slope, of the line
\bm{c}
is the value of the y-intercept

Outcomes

VCMNA283

Plot linear relationships on the Cartesian plane with and without the use of digital technologies

VCMNA284

Solve linear equations using algebraic and graphical techniques. Verify solutions by substitution

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