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2 Linear equations
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Australia
Year 10

2.06 Parallel and perpendicular lines

Lesson
Worksheet
Practice
Lesson

Parallel lines

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Parallel lines are lines that have the same gradient.

Shown here are two parallel lines that never cross and don't have any points in common.

Let's look at how we can identify parallel lines given their equations.

Equation formCharacteristic of parallel linesExamples
y=mx+c\text{Parallel lines have the same }m\text { value.}y=2x-1 \\ y=4+2x
ax+by+c=0\text{Parallel lines have the same value of } -\dfrac{a}{b}x+2y-3=0 \\ 2x+4y+1=0

For every straight line y=mx+c, there exist infinitely many lines parallel to it.

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Here is the line y=x and two more lines in the same family of parallel lines: y=x+3 and y=x-4.

Notice that they have the same gradient (m-value) but different x and y-intercepts (c-values).

Examples

Example 1

Which two lines are parallel to y=-3x+2?

A
y=3x
B
y=-\dfrac{2x}{3}+8
C
-3y-x=5
D
y=-10-3x
E
y+3x=7
Worked Solution
Create a strategy

Choose the two options with the same gradient, m, as the given line.

Apply the idea

In order for us to compare the gradients, we need to first make all the expressions into the form of y=mx+c. So we will rearrange options C and E.

For option C:

\displaystyle -3y-x\displaystyle =\displaystyle 5Write the equation
\displaystyle -3y\displaystyle =\displaystyle x+5Add x to both sides
\displaystyle \dfrac{-3y}{-3}\displaystyle =\displaystyle \dfrac{x+5}{-3}Divide both sides by -3
\displaystyle y\displaystyle =\displaystyle -\dfrac{1}{3}x - \dfrac{5}{3}Write in gradient intercept form

For option E:

\displaystyle y+3x\displaystyle =\displaystyle 7Write the equation
\displaystyle y\displaystyle =\displaystyle -3x+7Evaluate

The lines that have a gradient of m=-3 are y=-10-3x and y+3x=7.

So the lines in options D and E are parallel to y=-3x+2.

Idea summary

Parallel lines have the same gradient.

Horizontal and vertical lines

Horizontal lines are lines where the y-value is always the same.

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Let's look at the coordinates for A, B and C on this line.

A=\left(-8,4\right),\, B=\left(-2,4\right),\,C=\left(7,4\right)

All the y-coordinate are the same, y=4. Regardless of the x-value the y value is always 4.

The equation of this line is y=4

So if the equation of a straight line is y=c, then it will be a horizontal line passing through the point \left(0,c\right).

The x-axis itself is a horizontal line. The equation of the x-axis is y=0.

Vertical lines are lines where the x-value is always the same.

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Let's look at the coordinates for A, B and C on this line.

A=\left(-3,8\right), \,B=\left(-3,3\right), \,A=\left(-3,-3\right)

All the x-coordinate are the same, x=-3.

This means that regardless of the y-value the x value is always -3.

The equation of this line is x=-3

So if the equation of a straight line is x=c, then it will be a horizontal line passing through the point \left(c,0\right).

The y-axis itself is a horizontal line. The equation of the y-axis is x=0.

Examples

Example 2

Write down the equation of a line that is parallel to the x-axis and passes through \left(-10,2\right).

Worked Solution
Create a strategy

Determine whether the line is horizontal or vertical.

Apply the idea

A line that is parallel to the x-axis is a horizontal line which means it is of them form y=c. The y-coordinate of the given point it 2 so the equation of the line is: y=2

Idea summary

All horizontal lines are parallel to the x-axis and are of the form y=c.

Horizontal lines have a gradient of 0.

All vertical lines are parallel to the y-axis and are of the form x=c.

Vertical lines have an undefined gradient.

Perpendicular lines

Lines that meet at right angles (90\degree) are called perpendicular lines.

Exploration

Play with this applet to create pairs of perpendicular lines.

Fill in this table as you go.

Gradient of line 1m_{1}\\\\\\
Gradient of line 2m_{2}\\\\\\
Product of line 1 and line 2m_{1} \times m_{2}\\\\\\

What do you notice about the product of the gradients of lines 1 and 2? (The pair of perpendicular lines)

Loading interactive...

The product of the gradients of perpendicular lines will always be equal to -1.

Perpendicular lines have gradients whose product is equal to -1. So m_1 \times m_2 = -1.

We say that m_{1} is the negative reciprocal of m_{2}.

Negative reciprocal is a complex sounding term, but it just means two numbers that have opposite signs and are reciprocals of each other.

Here are some examples of negative reciprocals:

2 \ \text{ and } -\dfrac{1}{2} , \quad \quad \dfrac{3}{4} \ \text{ and } -\dfrac{4}{3}, \quad \quad \dfrac{3}{4} \ \text{ and } -\dfrac{4}{3}

Examples

Example 3

Find the equation of a line that is perpendicular to y=6x+10, and has the same y-intercept.

Give your answer in the form y=mx+c.

Worked Solution
Create a strategy

Find the gradient of the perpendicular line then substitute it into the equation y=mx+c.

Apply the idea

The gradient of y=6x+10 is 6. The negative reciprocal of 6 is -\dfrac{1}{6}.

So the gradient of the perpendicular line is m=-\dfrac{1}{6} and the y-intercept is c=10. So the equation of the line is: y=-\dfrac{1}{6}x+10

Idea summary

Two lines are perpendicular if their gradients are negative reciprocals of each other.

To test if lines are perpendicular multiply the gradients together. If the result is -1 then the lines are perpendicular.

Outcomes

ACMNA238

Solve problems involving parallel and perpendicular lines

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