To find the rule for a linear relation, which is often expressed in the form of y = mx + b (where m is the slope and b is the y-intercept), follow these straightforward steps:
Step-by-Step Guide
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Identify Two Points:
- Begin by identifying two distinct points on the graph of the linear relation. Let's call these points (x₁, y₁) and (x₂, y₂).
- For example, you might have the points (1, 3) and (2, 5).
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Calculate the Slope (m):
- Use the slope formula to find the value of m:
- m = (y₂ - y₁) / (x₂ - x₁)
- Using the example points (1, 3) and (2, 5):
- m = (5 - 3) / (2 - 1) = 2 / 1 = 2. The slope is 2.
- Simplify the Result: Make sure the slope is in its simplest form.
- Use the slope formula to find the value of m:
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Set Up the Function:
- Now that you have the slope (m), create a basic linear equation:
- f(x) = mx + b or y = mx + b
- Using our example slope:
- y = 2x + b
- Now that you have the slope (m), create a basic linear equation:
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Find the Y-intercept (b):
- Substitute either of your chosen points (x₁, y₁) or (x₂, y₂) into the equation y = mx + b. Let's use (1, 3) with the slope we found to be 2.
- 3 = 2(1) + b
- Solve for b:
- 3 = 2 + b
- b = 3 - 2
- b = 1
- Substitute either of your chosen points (x₁, y₁) or (x₂, y₂) into the equation y = mx + b. Let's use (1, 3) with the slope we found to be 2.
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Complete the Equation:
- Replace m and b in the equation y = mx + b with the values you found.
- Based on our example, the linear relation is:
- y = 2x + 1
Example Summary in Table Format
| Step | Action | Formula/Result | Example Values |
|---|---|---|---|
| 1 | Identify two points | (x₁, y₁) and (x₂, y₂) | (1, 3) and (2, 5) |
| 2 | Calculate the slope (m) | m = (y₂ - y₁) / (x₂ - x₁) | m = (5 - 3) / (2 - 1) = 2 |
| 3 | Set up the function | y = mx + b | y = 2x + b |
| 4 | Calculate the y-intercept (b) | y = mx + b (substitute a point) | 3 = 2(1) + b; b = 1 |
| 5 | Complete the equation | y = mx + b | y = 2x + 1 |
By following these steps, you can successfully determine the rule for any linear relationship given a graph or two points. This method is straightforward and applicable to any linear relation.
