11.10 Graphs of linear equations
We will now look at how to sketch a line directly from its equation, without needing to create a table of values first.
We have learnt that all linear relationships can be expressed in the form: $y=mx+c$y=mx+c, where $m$m is equal to the change in the $y$y-values for every increase in the $x$x-value by $1$1, and $c$c is the value of $y$y when $x=0$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 $x$x-axis and the $y$y-axis or they will pass through the origin, $\left(0,0\right)$(0,0).
Here are some examples:
We use the word intercept to refer to the point where the line crosses or intercepts with an axis.
- The $x$x-intercept is the point where the line crosses the $x$x-axis. The coordinates of the $x$x-intercept will always have a $y$y-coordinate of zero.
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The $y$y-intercept is the point where the line crosses the $y$y-axis. The coordinates of the $y$y-intercept will always have an $x$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 $x$x and $y$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 $x=0$x=0 or $y=0$y=0, which simplifies the work needed to solve.
The $y$y-intercept can be thought of as either the co-ordinate the point where the $y$y-axis is crossed, or simply the $y$y-value at this point (as the $x$x-value is by default $0$0).
Worked example
EXAMPLE 1
Find the $y$y-intercept for the straight line below:
The $y$y-intercept is $-6$−6, and the coordinates of the $y$y-intercept are $\left(0,-6\right)$(0,−6).
Practice Question
question 1
Consider the following graph.
Axis intersection at centre
Tick label format is fraction
The x major unit is $1$1
The x axis ends at $7$7
The x axis starts from $-7$−7
The x minor unit is $1$1
The y major unit is $1$1
The y axis ends at $7$7
The y axis starts from $-7$−7
The y minor unit is $1$1
The following elements are present:Function: 4 - (4*x)/3
State the $x$x-value of the $x$x-intercept.
State the $y$y-value of the $y$y-intercept.
question 2
Consider the linear equation $y=2x-4$y=2x−4.
What are the coordinates of the $y$y-intercept?
Give your answer in the form $\left(a,b\right)$(a,b).
What are the coordinates of the $x$x-intercept?
Give your answer in the form $\left(a,b\right)$(a,b).
Now, sketch the line $y=2x-4$y=2x−4:
Loading Graph...
The gradient of a line
The change in $y$y-values for every increase in the $x$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, $m$m, relates to the line $y=mx+c$y=mx+cas follows:
- A negative gradient ($m<0$m<0) means the line is decreasing
- A positive gradient ($m>0$m>0) means the line is increasing
- A zero gradient ($m=0$m=0) means the line is horizontal
- The higher the value of $m$m, the steeper the line.
Practice Question
question 3
What is the gradient $m$m of the line $y=9x+3$y=9x+3?
Gradient-intercept form of a straight line
Any straight line on the coordinate plane is defined entirely by its gradient and its $y$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: $y=mx+c$y=mx+c.
- $m$m is equal to the gradient, or slope, of the line.
- $c$c is the value of the $y$y-intercept.
We can use the applet below to see the effect of varying $m$m and $c$c on both the line and its equation.
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In algebra, any number written immediately in front of a variable, is called a coefficient. For example, in the term $3x$3x, the coefficient of $x$x is $3$3. Any number by itself is known as a constant term.
In the gradient-intercept form of a line, $y=mx+c$y=mx+c, the gradient, $m$m, is the coefficient of $x$x, and the $y$y-intercept, $c$c, is a constant term.
Practice Questions
question 4
Consider the line graph shown below:
The $x$x-value at the $y$y-intercept is $0$0.
What is the $y$y-value at this point?The equation of the line is $y=-2x+3$y=−2x+3.
This is of the form $y=mx+c$y=mx+c.
Which pronumeral represents the $y$y-intercept?
$y$y
A$m$m
B$x$x
C$c$c
D
Question 5
Consider the linear equation $y=2x+9$y=2x+9.
What are the values of the gradient $m$m and the $y$y-intercept $c$c?
$m$m $=$= $\editable{}$
$c$c $=$= $\editable{}$
General form of a straight line (Extended)
Straight lines can also be written in general form: $ax+by=d$ax+by=d, where $a>0$a>0 and $a,b$a,b and $d$d are integers.
Worked examples
example 2
Find the equation of the given line:
a) In gradient-intercept form.
b) In general form, $ax+by=d$ax+by=d.
a) Think: We can see the $y$y-intercept by looking where the line crosses the $y$y-axis. Then we can work out the gradient by considering the triangle formed by the line and the axes.
Do: The line crosses the $y$y-axis at $-6$−6, so the $y$y-intercept is: $c=-6$c=−6. The gradient can be found by considering the rise and run using the triangle formed by the line and the axes:
| $m$m | $=$= | $\frac{\text{rise}}{\text{run}}$riserun |
| $=$= | $\frac{6}{2}$62 | |
| $=$= | $3$3 |
Therefore, the equation of the line in gradient-intercept form, $y=mx+c$y=mx+c, is $y=3x-6$y=3x−6.
b) Think: We need to rearrange the equation from part (a) into the form $ax+by=d$ax+by=d.
Do:
| $y$y | $=$= | $3x-6$3x−6 |
The equation from (a) |
| $0$0 | $=$= | $3x-y-6$3x−y−6 |
Subtract $y$y from both sides. |
| $6$6 | $=$= | $3x-y$3x−y |
Add $6$6 to both sides. |
| $3x-y$3x−y | $=$= | $6$6 |
Swap the sides of the equation so it is in the correct form. |
Example 3
Rearrange the equation of the line $y=-\frac{2}{3}x+7$y=−23x+7 into general form.
Think: We must make sure that the coefficient of $x$x in the rearranged equation is positive, and that the coefficients and constant term are integers.
Do: We will need to get rid of the fraction and then move terms to the appropriate sides of the equal sign.
| $y$y | $=$= | $-\frac{2}{3}x+7$−23x+7 |
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| $3y$3y | $=$= | $-2x+21$−2x+21 |
Multiply both sides by three. |
| $2x+3y$2x+3y | $=$= | $21$21 |
Since the coefficient of x was negative, we need to move it to the LHS to become positive. |