Equations of a Straight Line

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How do I find the gradient of a straight line?

Find two points that the line passes through with coordinates (x₁, y₁) and (x₂, y₂)
The gradient between these two points is calculated by

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

- This is given in the formula booklet
The gradient of a straight line measures its slope
- A line with gradient 1 will go up 1 unit for every unit it goes to the right
- A line with gradient -2 will go down two units for every unit it goes to the right

What are the equations of a straight line?

$y = m x + c$
- This is the gradient-intercept form
- It clearly shows the gradient m and the y-intercept (0, c)
$y - y_{1} = m (x - x_{1})$
- This is the point-gradient form
- It clearly shows the gradient m and a point on the line (x₁, y₁)
$a x + b y + d = 0$
- This is the general form
- You can quickly get the x-intercept $(- \frac{d}{a}, 0)$ and y-intercept $(0, - \frac{d}{b})$

How do I find an equation of a straight line?

You will need the gradient
- If you are given two points then first find the gradient
It is easiest to start with the point-gradient form
- then rearrange into whatever form is required
  - multiplying both sides by any denominators will get rid of fractions
You can check your answer by using your GDC
- Graph your answer and check it goes through the point(s)
- If you have two points then you can enter these in the statistics mode and find the regression line $y = a x + b$

Exam Tip

A sketch of the graph of the straight line(s) can be helpful, even if not demanded by the question
- Use your GDC to plot them
Ensure you state equations of straight lines in the format required
- Usually $y = m x + c$ or $a x + b y + d = 0$
- Check whether coefficients need to be integers (they usually are for $a x + b y + d = 0$ )

Worked example

The line $l$ passes through the points $(- 2, 5)$ and $(6, - 7)$ .

Find the equation of $l$ , giving your answer in the form $a x + b y + d = 0$ where $a, b$ and $d$ are integers to be found.

2-1-1-ib-ai-sl-equation-of-a-line-we-solution

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Parallel Lines

How are the equations of parallel lines connected?

Parallel lines are always equidistant meaning they never intersect
Parallel lines have the same gradient
- If the gradient of line l₁is m₁ and gradient of line l₂is m₂then...
  - $m_{1} = m_{2} \Rightarrow l_{1} & l_{2} are parallel$
  - $l_{1} & l_{2} are parallel \Rightarrow m_{1} = m_{2}$
To determine if two lines are parallel:
- Rearrange into the gradient-intercept form $y = m x + c$
- Compare the coefficients of $x$
- If they are equal then the lines are parallel

Parallel & Perpendicular Gradients Notes Diagram 1

Worked example

The line $l$ passes through the point $(4, - 1)$ and is parallel to the line with equation $2 x - 5 y = 3$ .

Find the equation of $l$ , giving your answer in the form $y = m x + c$ .

2-1-1-ib-ai-sl-parallel-lines-we-solution

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Perpendicular Lines

How are the equations of perpendicular lines connected?

Perpendicular lines intersect at right angles
The gradients of two perpendicular lines are negative reciprocals
- If the gradient of line l₁is m₁ and gradient of line l₂is m₂then...
  - $m_{1} \times m_{2} = - 1 \Rightarrow l_{1} & l_{2} are perpendicular$
  - $l_{1} & l_{2} are perpendicular \Rightarrow m_{1} \times m_{2} = - 1$
To determine if two lines are perpendicular:
- Rearrange into the gradient-intercept form $y = m x + c$
- Compare the coefficients of $x$
- If their product is -1 then they are perpendicular
Be careful with horizontal and vertical lines
- $x = p$ and $y = q$ are perpendicular where p and q are constants

Parallel & Perpendicular Gradients Notes Diagram 3

Worked example

The line $l_{1}$ is given by the equation $3 x - 5 y = 7$ .

The line $l_{2}$ is given by the equation $y = \frac{1}{4} - \frac{5}{3} x$ .

Determine whether $l_{1}$ and $l_{2}$ are perpendicular. Give a reason for your answer.

2-1-1-ib-ai-sl-perpendicular-lines-we-solution

Equations of a Straight Line (DP IB Maths: AI SL)

Revision Note