Point-gradient equation

The point-gradient equation of a straight line is a formula used to write the equation of a line when you know the gradient $m$ and a point $(x_1,y_1)$ on the line. The general form of the point-gradient equation is:

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

Where:

• $m$ is the gradient (or slope) of the line.
• $(x_1,y_1)$ is a known point on the line.
• $x$ and $y$ are the variables representing any point on the line.

This formula is particularly useful when you know the gradient and a single point, as it allows you to quickly find the equation of the line.

1. Understanding the point-gradient equation:

The point-gradient equation is derived from the general equation $y=mx+c$, but it is written in a form that is more convenient when a specific point on the line is known. It expresses the relationship between the gradient and a point on the line. By using the known point, we can easily substitute its values to find the equation of the line.

2. Steps to use the point-gradient equation:

To find the equation of the line using the point-gradient equation, follow these steps:

• Write down the point $(x_1,y_1)$ and the gradient $m$ of the line.
• Substitute the values of $m$, $x_1$, and $y_1$ into the point-gradient equation: $$y-y_1=m(x-x_1)$$
• Simplify the equation as needed to get it into the desired form (e.g., slope-intercept form or general form).

3. Example: Using the point-gradient equation:

Let’s find the equation of a line with gradient $m=3$ that passes through the point $(2,5)$.

Step 1: Write the point-gradient equation:

Substitute $m=3$, $x_1=2$, and $y_1=5$ into the point-gradient equation: $$y-5=3(x-2)$$

Step 2: Simplify the equation:

Distribute the $3$ on the right-hand side: $$y-5=3x-6$$

Step 3: Isolate $y$:

Add $5$ to both sides to solve for $y$: $$y=3x-6+5$$ $$y=3x-1$$

4. The equation of the line:

The equation of the line is: $$y=3x-1$$

5. Check the solution:

To verify, substitute $x=2$ into the equation to ensure the point $(2,5)$ satisfies the equation: $$y=3(2)-1=6-1=5$$ This is correct, so the equation of the line is verified as $y=3x-1$.

6. Special cases to note:

• If the gradient $m=0$, the line is horizontal, and the equation becomes: $$y-y_1=0(x-x_1)\text{or simply}y=y_1$$
• If the gradient $m$ is undefined (i.e., the line is vertical), the equation becomes: $$x=x_1$$