5. Equation of a Line

The equation of a line in its general form is expressed as:

$y = mx + c$

Where:

• $m = \text{gradient}$
• $c = \text{y-intercept}$

The equation of a line is written in the form $y = mx + c$. Here, $y$ represents the output value of the line for a given $x$. The variable $m$ is known as the gradient (or slope), which tells us how steep the line is. The term $c$ represents the y-intercept, which is the point where the line crosses the y-axis. If you imagine a graph, the gradient indicates the angle of the line, while the y-intercept shows where the line starts on the y-axis when $x$ is 0.

Think of driving up a hill. The steepness of the hill relates to the gradient—steeper hills have a larger gradient, and more gradual slopes have a smaller gradient. The y-intercept is like where you would start your drive if you began at sea level. So if the line represents the path of your drive, $c$ tells you how high the starting point is on the vertical road.

To find the equation of a line given two points, we use the slope-point form:

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

When you have two points, say A$(x_1, y_1)$ and B$(x_2, y_2)$, you can calculate the gradient $m$ using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$. Once you have the gradient, you can substitute either point into the slope-point form of the equation, $y - y_1 = m(x - x_1)$, to express the line's equation. This format emphasizes how the line behaves starting from a specific point.

Imagine you're creating a road map based on two locations. The points A and B are like cities. The gradient is how steep the road is between those two cities. You can describe how to get from city A to city B effectively if you know the slope of the road and can start your journey from city A (point A) by saying how to adjust your path and elevation as you reach point B (your destination).

For example, to find the equation of the line through points A(1,2) and B(4,6):

1. Find the gradient: $m = \frac{6-2}{4-1} = \frac{4}{3}$
2. Use point A(1,2): $y - 2 = \frac{4}{3}(x - 1)$
3. Simplify: $y = \frac{4}{3}x + \left(2 - \frac{4}{3}\right) = \frac{4}{3}x + \frac{2}{3}$

To find the equation of a line using points A(1,2) and B(4,6), start by calculating the gradient using the formula for the slope. This gives you the steepness of the line. Next, plug in one of the points (in this case, A) into the slope-point form, which allows you to write the line's equation. By simplifying the expression, you ultimately derive the line's formula, which can be used to predict other points on the line.

Think of this process as building a bridge between two buildings (points). The gradient is akin to the incline of the bridge—how steep it will be. By starting at building A, you can calculate how to construct the rest of the bridge toward building B, adjusting based on its incline. The completed bridge equation shows you the path of your bridge, letting you know how straightforward the connection between the two locations is.

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