Equation of a Line Revision Notes for Leaving Cert Mathematics

Equation of a Line

What is the equation of a line?

The equation of a line is a mathematical statement that describes all the points that lie on that particular line. It shows the relationship between the x and y coordinates of any point on the line.

When we look at a line on a coordinate plane, we can find a pattern in the coordinates. For example, if we examine the line shown above, we notice that for all points on the line, when we add the x and y coordinates together, we always get the same number (5 in this case). This gives us the equation x + y = 5.

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The key insight here is that every point on a line satisfies the same mathematical relationship between its x and y coordinates. This consistent relationship is what we call the equation of the line.

The point-slope form

The most useful method for finding the equation of a line is called the point-slope form. This method works when you know:

One point on the line: (x₁, y₁)
The slope of the line: m

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Point-slope form formula:

$y - y_1 = m(x - x_1)$

This is the most important formula for finding line equations and should be memorised.

This formula comes from the definition of slope. Since slope equals rise over run, we can write:

$m = \frac{y - y_1}{x - x_1}$

When we multiply both sides by (x - x₁), we get the point-slope form.

Method 1: finding the equation when given a slope and a point

Step 1: Identify the given information

Slope (m)
One point (x₁, y₁)

Step 2: Substitute into the point-slope formula

Use y - y₁ = m(x - x₁)

Step 3: Simplify and rearrange

Expand the brackets
Collect all terms on one side
Write in the form axe + by + c = 0

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Worked Example 1: Finding equation with given slope and point

Find the equation of the line containing the point (-3, 2) with slope ²⁄₃.

Solution:

Given: m = ²⁄₃ and point (-3, 2), so x₁ = -3 and y₁ = 2
Using the formula: y - y₁ = m(x - x₁)
Substitute: y - 2 = ²⁄₃(x + 3)
Expand: y - 2 = ²ˣ⁄₃ + ⁶⁄₃
Simplify: y - 2 = ²ˣ⁄₃ + 2
Multiply through by 3: 3y - 6 = 2x + 6
Rearrange: 2x - 3y + 12 = 0

Therefore, the equation is 2x - 3y + 12 = 0

Method 2: finding the equation when given two points

When you have two points on a line, you need to follow a systematic approach to find the slope first, then apply the point-slope form.

Step 1: Find the slope using the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$

Step 2: Use either point with the point-slope form

Choose either of the two points as (x₁, y₁)
Apply y - y₁ = m(x - x₁)

Step 3: Simplify to get the final equation

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Worked Example 2: Finding equation with two points

Find the equation of the line containing the points (-2, 3) and (3, 1).

Solution:

Step 1: Calculate the slope
- m = (y₂ - y₁)/(x₂ - x₁) = (1 - 3)/(3 - (-2)) = -2/5
Step 2: Use the slope and point (-2, 3)
- y - y₁ = m(x - x₁)
- y - 3 = -²⁄₅(x + 2)
- y - 3 = -²ˣ⁄₅ - ⁴⁄₅
Step 3: Simplify
- Multiply by 5: 5y - 15 = -2x - 4
- Rearrange: 2x + 5y - 11 = 0

Therefore, the equation is 2x + 5y - 11 = 0

Key relationships with slope

Understanding slope relationships is crucial for line equations and will help you solve more complex problems involving parallel and perpendicular lines.

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Important Slope Relationships:

Parallel lines have the same slope
Perpendicular lines have slopes that are negative reciprocals of each other
If one line has slope m, a perpendicular line has slope -1/m

Different forms of line equations

Line equations can be written in several forms, each serving different purposes:

Point-slope form: y - y₁ = m(x - x₁)
General form: axe + by + c = 0
Slope-intercept form: y = mx + c (where c is the y-intercept)

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Most exam questions require the answer in general form (axe + by + c = 0). Always check the question requirements to ensure you provide your answer in the correct format.

Exam tips

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Essential Exam Strategies:

Always check your arithmetic when expanding brackets
Remember to collect like terms carefully
When finding slope, be careful with negative signs
The point-slope formula works with any point on the line
Always simplify your final answer to the form axe + by + c = 0

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Key Points to Remember:

The point-slope form y - y₁ = m(x - x₁) is the key formula for finding line equations
To find an equation with two points, first calculate the slope using m = (y₂ - y₁)/(x₂ - x₁)
Parallel lines have equal slopes, while perpendicular lines have slopes that multiply to give -1
Always simplify your final equation to the general form axe + by + c = 0
Double-check
your arithmetic, especially when dealing with negative numbers and fractions

Equation of a Line (Leaving Cert Mathematics): Revision Notes