3 Linear equations

Lesson

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

Equation form | Characteristic of parallel lines | Examples |
---|---|---|

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.

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

Idea summary

Parallel lines have the same gradient.

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

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.

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.

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

Worked Solution

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.

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

Play with this applet to create pairs of perpendicular lines.

Fill in this table as you go.

Gradient of line 1 | m_{1} | \\ | \\ | \\ |
---|---|---|---|---|

Gradient of line 2 | m_{2} | \\ | \\ | \\ |

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

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

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}

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

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.

Solve problems involving gradients of parallel and perpendicular lines