Equation of a Line Revision Notes for Junior Cert Mathematics

Equation of a Line

In this section, we will explore what the equation of a line is and how to find it using two important formulas. We will also learn how to use the equation of a line to find points on the line. Don't worry—we'll take it step by step to make everything as clear as possible.

What is the Equation of a Line?

The equation of a line is a mathematical rule that describes every point on the line. It's like a "recipe" that tells you how to find any point along the line. By knowing the equation, you can figure out important details, like how steep the line is and where it crosses the axes.

There are two main formulas for finding the equation of a line:

Point-Slope Formula: Useful when you know a point on the line and the slope.
Slope-Intercept Formula: Shows the slope of the line and where it crosses the y-axis.

1. Point-Slope Formula

The Point-Slope Formula is used when you know:

The slope of the line (how steep it is).
A point on the line (a specific spot the line passes through). The formula looks like this:

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

Here's what each part means:

$m$ : The slope of the line (how steep it is).
$(x_1, y_1)$ : A point on the line, where $x_1$ is the x-coordinate and $y_1$ is the y-coordinate.

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Example 1: Using the Point-Slope Formula

Let's say you know the slope of a line is $2$ , and the line passes through the point $(3, 4)$ . How do we find the equation of the line?

Step 1: Identify what you know:

The slope is $m = 2$ .
The point on the line is $(x_1, y_1) = (3, 4)$ .

Step 2: Substitute the numbers into the formula:

$y - (4) = 2(x - (3))$

Step 3: Simplify the equation: Now, let's simplify it step by step to make it easier to use.

Distribute the 2 on the right side: $y - 4 = 2x - 6$
Then, move everything to one side of the equation to fully simplify it: $2x - y - 2 = 0$ This is the fully simplified equation of the line.

2. Slope-Intercept Formula

The Slope-Intercept Formula is used when you know:

The slope of the line.
The y-intercept (where the line crosses the $y-axis$ ). The formula looks like this:

$y = mx + c$

Here's what each part means:

$m$ : The slope of the line (how steep it is).
$c$ : The y-intercept (where the line crosses the $y-axis$ ).

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Example 2: Using the Slope-Intercept Formula

Let's say you know that the slope of a line is $3$ , and it crosses the $y-axis$ at the point $(0, -5)$ . How do we find the equation of the line?

Step 1: Identify what you know:

The slope is $m = 3$ .
The y-intercept is $c = -5$ .

Step 2: Substitute the numbers into the formula:

$y = 3x - 5$

Step 3: Simplify the equation: To fully simplify it, move everything to one side of the equation:

$3x - y - 5 = 0$

This is the fully simplified equation of the line.

How to Use the Equation of a Line to Find Points

Once you have the equation of a line, you can find any point on that line by plugging in a value for $x$ and solving for $y$ . This tells you what the corresponding $y$ value is when you choose a particular $x$ value.

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Example 3: Finding Points on the Line

Let's use the equation we found earlier: $2x - y - 2 = 0$ . How can we find points on this line?

Step 1: Choose a value for $x$ :

Let's say $x = 1$ .

Step 2: Substitute this value into the equation:

$2(1) - y - 2 = 0$

Step 3: Solve for $y$ :

Simplify the equation: $2 - y - 2 = 0$
Combine like terms: $-y = 0$
Multiply both sides by $-1$ : $y = 0$

So, when $x = 1, y = 0$ . This means the point $(1, 0)$ is on the line.

You can repeat this process with different values of $x$ to find as many points on the line as you need.

Key Tips for Success

Always Label: When you're given a point, label it clearly as $(x_1, y_1)$ . This will help you avoid mistakes when substituting into the formulas.
Substitute Carefully: When you plug numbers into the formulas, use brackets, especially if the numbers are negative. This helps prevent mistakes.
Simplify the Equation: After using the point-slope formula, try to simplify the equation by moving everything to one side. This makes it easier to work with.
Practice Finding Points: Use the equation of the line to find different points on the line. This helps you see how the line behaves on the graph. With practice, finding and using the equation of a line will become easier, and you'll be able to describe any line on the Cartesian Plane with confidence!

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