# Lesson 1.2 Lines and their equations

This is a free lesson. We trust you enjoy it!

## The equation of a curve

Watch this video, which explains the idea of the equation of a curve.

## The equation of a curve

1. The equation of a curve indicates the relationship between the  $$x$$- and  $$y$$-coordinates of all points on the curve. Each type of function/relation has a characteristic shape.
2. A curve (or line) is the set of all points whose coordinates satisfy the equation of the curve (or line).
3. If a point lies on a curve, its coordinates will satisfy the equation of the curve. If a point does not lie on the curve, its coordinates will not satisfy the equation of the curve.

## Straight lines

A straight line is a "curve" with a constant gradient — think about why this is the case! The following video explains how we derive two equations for a straight line. You don't have to know how to derive these formulae, but you must know them well, and be able to use them. You will do so hundreds of times before completing this course, and you will definitely have to use at least one of them in the exam!

What about horizontal and vertical lines? Can you suggest what their equations will look like?

## The equation of a straight line

The equation of a straight line with gradient $$m$$ and passing through point $$A$$ with coordinates $$x_A$$ and $$y_A$$ is given by

$$y-y_A = m(x-x_A)$$

The equation of a straight line can be written in the form

$$y=mx+c$$

where  $$m$$ is the gradient of the line and  $$c$$ is the  $$y$$-intercept.

The equation of a horizontal line has the form  $$y=k$$,

and the equation of a vertical line has the form  $$x=k$$,

where  $$k$$ is a constant.

The following videos show you how to find the equation of a straight line when you are given either the gradient and one point on the line, or two points on the line.

Read sections 1.7 and 1.8 in your text book and make sure you understand them. The following video shows you how to find the point of intersection of

$$x+5y=22$$  and  $$3x+2y=14$$.

(See Exercise 1A no. 11(e).)

## Weight lifting

Complete the following exercises from your text book:

Exercise 1B

• Question 1 b, d, f, h.
• Question 2

Hints:

• For questions 5 - 10, note that parallel lines have the same gradient. In questions 7 & 8, you must first work out the gradient of the line through the last two points given; this allows you to work out the equation of the line through the first point.
• For question 11 you must solve simultaneous equations (see section 1.8 in the text book).