Finding the Equation of a Line: Double Elimination Method

In the world of mathematics, lines are fundamental building blocks. They represent relationships between variables, and understanding their equations is crucial for solving various problems. One common task is finding the equation of a line when given two points. This is where the double elimination method comes in handy. It's a straightforward technique that allows us to determine the equation of a line systematically.

Understanding the Equation of a Line

Before diving into the double elimination method, let's review the standard form of a linear equation:

y = mx + b

• y represents the dependent variable (usually plotted on the vertical axis).
• x represents the independent variable (usually plotted on the horizontal axis).
• m represents the slope of the line, which indicates its steepness.
• b represents the y-intercept, the point where the line crosses the y-axis.

The Double Elimination Method: A Step-by-Step Guide

The double elimination method leverages the fact that each point on a line satisfies its equation. Here's how it works:

1. Step 1: Identify the Two Points

Let's say you are given two points, (x₁, y₁) and (x₂, y₂).

2. Step 2: Substitute the Points into the Equation

Substitute the coordinates of each point into the equation y = mx + b:

• For point (x₁, y₁): y₁ = mx₁ + b
• For point (x₂, y₂): y₂ = mx₂ + b
3. Step 3: Create a System of Equations

Now you have two equations with two unknowns (m and b):

• Equation 1: y₁ = mx₁ + b
• Equation 2: y₂ = mx₂ + b
4. Step 4: Eliminate 'b'

To eliminate 'b', subtract Equation 1 from Equation 2:

y₂ - y₁ = (mx₂ + b) - (mx₁ + b)

Simplifying, we get:

y₂ - y₁ = mx₂ - mx₁

5. Step 5: Solve for 'm'

Factor out 'm' from the right side of the equation:

y₂ - y₁ = m(x₂ - x₁)

Isolate 'm' by dividing both sides by (x₂ - x₁):

m = (y₂ - y₁) / (x₂ - x₁)

6. Step 6: Substitute 'm' to Find 'b'

Substitute the value of 'm' you just calculated back into either Equation 1 or Equation 2. Let's use Equation 1:

y₁ = [(y₂ - y₁) / (x₂ - x₁)]x₁ + b

Solve for 'b' by simplifying and isolating it.

7. Step 7: Write the Equation of the Line

You now have the values for both 'm' and 'b'. Substitute them into the standard form of the equation:

y = mx + b

Example: Finding the Equation of a Line

Let's say we have two points: (2, 3) and (5, 9).

1. Step 1: Identify the Two Points

(x₁, y₁) = (2, 3)

(x₂, y₂) = (5, 9)

2. Step 2: Substitute the Points into the Equation

Equation 1: 3 = 2m + b

Equation 2: 9 = 5m + b

3. Step 3: Create a System of Equations

Equation 1: 3 = 2m + b

Equation 2: 9 = 5m + b

4. Step 4: Eliminate 'b'

9 - 3 = (5m + b) - (2m + b)

6 = 3m

5. Step 5: Solve for 'm'

m = 6 / 3 = 2

6. Step 6: Substitute 'm' to Find 'b'

3 = (2)(2) + b

3 = 4 + b

b = -1

7. Step 7: Write the Equation of the Line

y = 2x - 1

Conclusion

The double elimination method provides a systematic and efficient way to determine the equation of a line given two points. By following these steps, you can confidently solve for the slope and y-intercept, ultimately finding the equation that represents the relationship between the variables.