5.2. Equation Using Two Points

1. Find gradient:

   $m = \frac{y_2 - y_1}{x_2 - x_1}$

   For points 𝐴(1,2) and 𝐵(4,6):

   $m = \frac{6 - 2}{4 - 1} = \frac{4}{3}$

To find the gradient of a line between two points, we calculate the difference in their y-coordinates and divide it by the difference in their x-coordinates. Here, we have two points A(1, 2) and B(4, 6). After substituting into the gradient formula, we calculate:

1. Subtract the y-coordinates: $6 - 2 = 4$
2. Subtract the x-coordinates: $4 - 1 = 3$
3. Divide the results: $m = \frac{4}{3}$. This means that for every 3 units we move to the right, we move up 4 units along the line.

Imagine you are climbing a hill. If you move 3 steps sideways and go up 4 steps at the same time, the slope of that hill is $\frac{4}{3}$. This steepness describes how steep the path is.

2. Use point 𝐴(1,2):

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

   Substitute the values:

   $y - 2 = \frac{4}{3}(x - 1)$

After finding the gradient, we use the point-slope form of the equation of the line. The formula is: $y - y_1 = m(x - x_1)$. Here, we take the coordinates of point A, which are (1, 2), and substitute our gradient. Therefore, we convert the coordinates and gradient into the line's equation as follows: $y - 2 = \frac{4}{3}(x - 1)$. This equation expresses the relationship between y and x, indicating how y changes based on x.

Think of this equation as giving directions. If you start at point (1, 2), moving according to the slope of $\frac{4}{3}$ tells you how to move up on a map depending on your movement left or right.

3. Simplify:

   $y = \frac{4}{3}x + (2 - \frac{4}{3}) = \frac{4}{3}x + \frac{2}{3}$

Now, we need to rearrange the equation found in the last step into the standard slope-intercept form, which is $y = mx + c$. We start with $y - 2 = \frac{4}{3}(x - 1)$ and distribute the gradient across the bracket. After simplification, we ultimately reach: $y = \frac{4}{3}x + \frac{2}{3}$. Here, the coefficient of x ($\frac{4}{3}$) is the gradient and the last term ($\frac{2}{3}$) is the y-intercept.

You can think of simplifying the equation like cooking from a recipe. Initially, you combine ingredients (your initial equation), but then you mix and bake (simplify) until you have your final dish (the simplified line equation). This is crucial because it clearly identifies how 'y' relates to 'x'.

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