Conquering Linear Equations: Crafting $y = mx + c$

In mathematics, the story of a straight line is captured by the equation $y = mx + c$. Here’s how you can find these crucial components and craft the equation of your line.

 

 

Determining the Gradient ($m$)

The gradient or slope of a line measures its steepness and direction. If you know two points on the line, let’s say $(x_1, y_1)$ and $(x_2, y_2)$, you can calculate the gradient ($m$) using the formula:

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

This formula gives you a numerical value representing how many units the line goes up (or down) for each unit it goes to the right.

Finding the Y-intercept ($c$)

The y-intercept is the point where the line crosses the y-axis. To find $c$ when you know the gradient and at least one point on the line ($x$, $y$), you can substitute the values into the equation $y = mx + c$ and solve for $c$.

Piecing Together $y = mx + c$

Once you have $m$ and $c$, you can write the equation of the line. This equation tells you the vertical position ($y$) of any point on the line, based on its horizontal position ($x$), its steepness ($m$), and where it intersects the y-axis ($c$).

 

Example

Given points $(4, 7)$ and $(9, 17)$ on a line, find the equation of the line.

 

 

Solution

Step 1: First find the gradient of the line $$m = \frac{17 - 7}{9 - 4}  \to  m = \frac{10}{5}  = 2  \therefore m = 2$$

Step 2: Now use the gradient and the general equation to find the $+c$. Pick a coordinate on the line and substitue in to the equation and solve for c i.e $$y = 2x + c \\  \to \text{(I will use first coordinate)} 7 = 2(4) + c \\ 7 = 8 + c \to 7-8 = c \to -1 = c \therefore c = -1 \\ \text{Solution}  y = 2x - 1$$