Linear Equations and Relationships

TL;DR

Linear equations describe straight-line relationships between variables, helping you predict one value given another. They typically look like y = mx + c, where m is the steepness (gradient) and c is where the line crosses the y-axis (y-intercept). Understanding these parts lets you graph lines, find missing values, and solve real-world problems.

1. The Mental Model

Think of a linear relationship like a recipe: if you double one ingredient, you double the output. It's a steady, predictable change. When plotted on a graph, this steady change always forms a straight line.

2. The Core Material

Linear equations are super useful for showing how two things are connected in a simple, predictable way. The most common form you'll see is:

y = mx + c

Let's break down what each part means:

• y: This is your dependent variable. Its value depends on x. Think of it as the "output".
• x: This is your independent variable. You choose or know its value, and it determines y. Think of it as the "input".
• m: This is the gradient (or slope). It tells you how steep the line is and in which direction it's going (uphill or downhill). A positive m means the line goes up from left to right, and a negative m means it goes down.
  • How to find m: m = (change in y) / (change in x) or m = (y2 - y1) / (x2 - x1).
• c: This is the y-intercept. It's the point where the line crosses the y-axis. At this point, x is always 0.

Understanding the Gradient (m)

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The gradient tells you the rate of change. For example, if y is the cost of a taxi ride and x is the distance, m would be the cost per kilometre.

• If m = 2, for every 1 unit x increases, y increases by 2 units.
• If m = -3, for every 1 unit x increases, y decreases by 3 units.
• If m = 0, the line is horizontal (no change in y as x changes).

Understanding the y-intercept (c)

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The y-intercept c is your starting point or base value. In the taxi example, c could be the fixed booking fee, even if you travel 0 km.

Graphing a Linear Equation

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To graph y = mx + c, you really only need two points, but plotting three is a good way to check your work.

1. Plot the y-intercept (c): This is the easiest point to find. Just mark (0, c) on your graph.
2. Use the gradient (m) to find another point: From the y-intercept, use m = rise / run. If m = 2/1, go up 2 units and right 1 unit. If m = -3/1, go down 3 units and right 1 unit.
3. Draw the line: Connect your points with a straight line, extending it with arrows to show it goes on forever.

Or, you can just pick any two x values, calculate their y values, plot those two (x, y) points, and draw the line.

Rearranging Equations

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Sometimes, equations aren't in the y = mx + c form. You'll need to use your algebra skills to rearrange them. The goal is to get y by itself on one side.

Example: 2x + y = 5
To get y alone, subtract 2x from both sides:
y = -2x + 5
Now you can see m = -2 and c = 5.

Here's a flowchart to help you think about graphing:

graph TD
    A["Start with an equation"] --> B{"Is it in y = mx + c form?"}
    B -- No --> C["Rearrange to y = mx + c"]
    B -- Yes --> D["Identify 'm' (gradient) and 'c' (y-intercept)"]
    D --> E["Plot the y-intercept (0, c)"]
    E --> F{"Use 'm' (rise/run) or pick another x value?"}
    F -- Use 'm' --> G["From (0,c), move 'rise' units vertically and 'run' units horizontally to find a second point"]
    F -- Pick x value --> H["Choose a value for x (e.g., x=1). Substitute into y = mx + c to find corresponding y value. Plot (x,y)"]
    G --> I["Draw a straight line through the two points"]
    H --> I
    I --> J["Extend line with arrows"]
    J --> K["End"]

3. Worked Example

Let's work with the equation: 3x - y = 6

1. Rearrange to y = mx + c form:
   3x - y = 6
   Subtract 3x from both sides:
   -y = -3x + 6
   Multiply the entire equation by -1 (or divide by -1) to make y positive:
   y = 3x - 6

2. Identify m and c:
   From y = 3x - 6, we have:

   • m = 3 (The gradient is 3)
   • c = -6 (The y-intercept is -6)

3. Find points to plot:

   • Y-intercept: (0, -6)
   • Using gradient m = 3 (which is 3/1): From (0, -6), go up 3 units and right 1 unit. This gives you the point (1, -3).
   • Alternative (picking another x): Let x = 2.
     y = 3(2) - 6
y = 6 - 6
y = 0
     So, another point is (2, 0). This is where the line crosses the x-axis (the x-intercept).

4. Plot these points (0, -6), (1, -3), (2, 0) and draw a straight line through them.

4. Key Takeaways

• The form y = mx + c is the most important for understanding linear equations.
• m represents the gradient or steepness, showing how much y changes for each unit change in x.
• c represents the y-intercept, which is where the line crosses the y-axis (when x = 0).
• To graph a linear equation, you only need two points, but using the y-intercept and gradient is often quickest.
• Always check if your equation is in y = mx + c form before identifying m and c.

Common Mistakes to Avoid:
- Forgetting to change the signs when moving terms across the equals sign during rearrangement.
- Confusing m and c – m is always with x, c is always the constant term.
- Miscalculating the gradient, especially with negative numbers or fractions.
- Drawing a curve instead of a straight line – linear equations always make straight lines!

5. Now Try It

You've got this! Take the equation 5x + 2y = 10.

1. Rearrange it into the y = mx + c form.
2. Identify the gradient (m) and the y-intercept (c).
3. Find three points that lie on this line (e.g., the y-intercept, the x-intercept, and one other point).

Success looks like: You have the correct y = mx + c form, clearly stated m and c values, and three correct coordinate pairs that satisfy the equation.