Introduction to the Straight Line

In Further Pure Mathematics, the straight line is one of the most fundamental shapes we study. At its heart, a straight line represents a constant relationship between two variables, usually \(x\) and \(y\). Whether you are predicting trends in physics or finding the shortest path between two points, understanding how to write and manipulate the equation of a line is a vital skill.

Don't worry if algebra feels like a puzzle at first! By the end of these notes, you will be able to look at a line and describe it perfectly using math. This chapter focuses specifically on the different forms of a line's equation and how to use them.

Note: For help on finding the distance between points or calculating the gradient itself, see the "Distance and Gradient" chapter.

1. The Slope-Intercept Form: \(y = mx + c\)

This is likely the version of a line you are most familiar with. It is very useful because it tells you exactly how the line looks just by glancing at it.

  • \(m\) represents the gradient (the "steepness" or "slope").
  • \(c\) represents the y-intercept (where the line crosses the vertical \(y\)-axis).

How to use it:

If you have a gradient of \(3\) and the line crosses the \(y\)-axis at \((0, -2)\), your equation is simply \(y = 3x - 2\).

Analogy: The Mountain Climber

Imagine a mountain climber. The gradient \(m\) tells them how steep the climb is (a high \(m\) is a cliff, a negative \(m\) is a downhill slope). The intercept \(c\) tells them their starting altitude when they begin at horizontal position zero.

2. The Point-Gradient Form: \(y - y_1 = m(x - x_1)\)

This form is often the "secret weapon" for Further Pure Math students. It is much faster to use than \(y = mx + c\) when you are given a specific point and a gradient.

  • \(m\) is the gradient.
  • \((x_1, y_1)\) is a specific point that the line passes through.

Step-by-Step: Finding an equation

Suppose a line has a gradient of \(4\) and passes through the point \((2, 5)\).

  1. Identify your values: \(m = 4\), \(x_1 = 2\), and \(y_1 = 5\).
  2. Plug them into the formula: \(y - 5 = 4(x - 2)\).
  3. Expand the brackets: \(y - 5 = 4x - 8\).
  4. Simplify to get it into the form the question asks for: \(y = 4x - 3\).

Quick Tip: This form is especially useful in Calculus when finding the equations of tangents and normals!

3. The General Form: \(ax + by = c\)

Sometimes, you will see equations written like this: \(3x + 2y = 12\). This is called the general form. While it doesn't show the gradient immediately, it is very "tidy" and often used in exam questions.

Decoding the General Form

To find the gradient and intercept from this form, just rearrange it to make \(y\) the subject (get \(y\) on its own):

\(3x + 2y = 12\)
\(2y = -3x + 12\)
\(y = -\frac{3}{2}x + 6\)

Now we can see clearly that the gradient \(m = -\frac{3}{2}\) and the y-intercept \(c = 6\).

4. Finding an Equation from Two Points

If a question gives you two points, say \(A(x_1, y_1)\) and \(B(x_2, y_2)\), and asks for the equation, follow these two steps:

  1. Find the Gradient: Use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
  2. Use the Point-Gradient Form: Pick either point and use it with your new gradient in \(y - y_1 = m(x - x_1)\).
Example:

Find the equation of the line passing through \((1, 4)\) and \((3, 10)\).

1. \(m = \frac{10 - 4}{3 - 1} = \frac{6}{2} = 3\).
2. Using point \((1, 4)\): \(y - 4 = 3(x - 1)\).
3. \(y - 4 = 3x - 3 \implies y = 3x + 1\).

Did you know? A vertical line (straight up and down) has no \(y\) in its equation. It is just written as \(x = k\), where \(k\) is the value it passes through on the x-axis. Its gradient is undefined!

Common Mistakes to Avoid

  • Sign Errors: When using \(y - y_1 = m(x - x_1)\), if your point has a negative coordinate, remember that subtracting a negative makes a positive. For example, if \(x_1 = -3\), the bracket becomes \((x + 3)\).
  • Mixing up \(x\) and \(y\): Always remember that the gradient is "rise over run" (\(y\) change divided by \(x\) change).
  • Forgetting to simplify: Exams often ask for the equation in a specific format (like \(ax + by + c = 0\)). Always check the question's instructions at the very end!

Quick Review: Key Takeaways

1. Slope-Intercept: \(y = mx + c\) (Best for graphing).
2. Point-Gradient: \(y - y_1 = m(x - x_1)\) (Best for building an equation).
3. General Form: \(ax + by = c\) (Best for a tidy final answer).
4. Rearranging: To find the gradient of any line, just solve for \(y\).