Equation of a Straight Line

1. The Anatomy of a Line Equation: y = mx + c

Every straight line on a coordinate grid can be described by the equation:

$y=mx+c$

You are already very familiar with the first part of this equation:

• $m$ is the gradient of the line, which you learned to calculate in the last lesson. It tells us the steepness of the line.
• $c$ is the y-intercept. This is the y-coordinate of the point where the line crosses the vertical y-axis.

To define a unique straight line, we need to find the specific values of $m$ and $c$. Our job today is to learn the systematic process for doing this.

2. The Core Method: Finding the Equation from Two Points

This is the most common type of question. If you are given any two points on a line, say $(x_1,y_1)$ and $(x_2,y_2)$, you can find its equation using a clear, two-step process.

The Two-Step Process:

1. Find the gradient ($m$): Use the formula $m=\frac{y_2-y_1}{x_2-x_1}$.
2. Find the y-intercept ($c$): Substitute the gradient ($m$) you just found, and the coordinates of one of the points ($x$ and $y$), into the equation $y=mx+c$. Then, solve the resulting simple linear equation to find $c$.

To avoid the small arithmetic errors you've noted as a challenge, it's often wise to pick the point with simpler coordinates (e.g., positive numbers, smaller numbers, or zeros) for the substitution in Step 2.

The following video provides an excellent, clear walkthrough of this exact process.

3. The Simpler Case: Given Gradient and One Point

Sometimes, the examiner makes your life easier by giving you the gradient directly. In this case, you can skip Step 1 and go straight to Step 2.

Method:

1. Take the given gradient $m$.
2. Substitute $m$ and the coordinates of the given point $(x,y)$ into $y=mx+c$.
3. Solve for $c$ and write the final equation.

This is a more direct application of the same principle. Let's look at a quick written example to solidify this.

4. Practice: The Error-Loop Method

You've learned the method; now it's time to apply it under exam-like conditions. This is where you'll sharpen your skills, improve your speed, and catch any recurring errors in your process.

Conclusion

Excellent work today. You have mastered a crucial algebraic skill that is central to the entire coordinate geometry topic. By internalizing the two-step process for finding a line's equation, you have a reliable strategy for a wide range of exam questions.

Key Takeaways:

• The equation of a straight line is always in the form $y=mx+c$.
• The core method is a two-step process: 1. Find the gradient $m$, 2. Substitute a point to find the y-intercept $c$.
• If the gradient is given, you only need to perform the second step.
• Methodical working and careful checking are the best ways to avoid simple arithmetic errors, especially with negative numbers.

Preview of the Next Lesson:
We will now extend this skill to solve problems involving parallel and perpendicular lines. In the next lesson, you will learn the simple but powerful rules connecting their gradients:

• Parallel lines have equal gradients.
• The gradients of perpendicular lines multiply to give -1.
  This will unlock another set of common and high-value exam questions.