Finding the y-intercept of a line, given just two points, might seem tricky, but it's a straightforward process using the equation of a line. The y-intercept is where the line crosses the y-axis, meaning the x-coordinate is zero. This guide will walk you through the steps, explaining the concepts clearly and providing examples.
Understanding the Equation of a Line
Before we dive into finding the y-intercept, let's refresh our understanding of the equation of a line. The most common form is the slope-intercept form:
y = mx + b
Where:
- y and x represent the coordinates of any point on the line.
- m is the slope of the line (how steep it is).
- b is the y-intercept (the y-coordinate where the line crosses the y-axis).
Calculating the Slope (m)
The first step is to find the slope (m) using the two points you're given. Let's say your two points are (x₁, y₁) and (x₂, y₂). The formula for the slope is:
m = (y₂ - y₁) / (x₂ - x₁)
Example: Let's say our two points are (2, 4) and (6, 10).
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Identify x₁ y₁, x₂ and y₂:
- x₁ = 2
- y₁ = 4
- x₂ = 6
- y₂ = 10
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Apply the slope formula:
m = (10 - 4) / (6 - 2) = 6 / 4 = 3/2
Therefore, the slope (m) is 3/2.
Finding the Y-Intercept (b)
Now that we have the slope, we can use one of the given points and the slope-intercept form (y = mx + b) to solve for the y-intercept (b).
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Substitute: Choose either of your original points (let's use (2, 4) for this example). Substitute the values of x, y, and m into the equation:
4 = (3/2)(2) + b
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Solve for b:
4 = 3 + b b = 4 - 3 b = 1
Therefore, the y-intercept (b) is 1.
Putting it All Together: The Equation of the Line
Now we have both the slope (m = 3/2) and the y-intercept (b = 1), so we can write the complete equation of the line:
y = (3/2)x + 1
Example 2: Working with Negative Numbers
Let's try another example with negative numbers to illustrate the process further. Suppose the points are (-1, -3) and (2, 3).
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Calculate the slope:
m = (3 - (-3)) / (2 - (-1)) = 6 / 3 = 2
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Find the y-intercept: Using point (2, 3) and the slope m = 2:
3 = 2(2) + b 3 = 4 + b b = -1
Therefore, the y-intercept is -1, and the equation of the line is y = 2x - 1.
Key Takeaways
Finding the y-intercept given two points is a fundamental skill in algebra. By mastering the steps outlined above – calculating the slope and then using it to solve for the y-intercept – you can confidently determine the equation of any straight line. Remember to always double-check your calculations to ensure accuracy. Practice with various examples to build your proficiency!
