How To Find Yintercept With Two Points

Finding the y-intercept of a line given two points is a fundamental concept in algebra and geometry. The y-intercept is the point at which the line crosses the y-axis, and it’s an essential component of the slope-intercept form of a line (y = mx + b), where m is the slope of the line and b is the y-intercept.

To find the y-intercept with two points, you can follow a step-by-step approach that involves calculating the slope of the line first and then using one of the points along with the slope to find the y-intercept.

Step 1: Understand the Formula for Slope

The slope (m) of a line passing through two points ((x_1, y_1)) and ((x_2, y_2)) can be calculated using the formula: [m = \frac{y_2 - y_1}{x_2 - x_1}]

Step 2: Calculate the Slope

Given two points, for example, ((3, 4)) and ((6, 8)), you can substitute these values into the slope formula: [m = \frac{8 - 4}{6 - 3} = \frac{4}{3}] So, the slope of the line is (⁴⁄₃).

Step 3: Use the Slope-Intercept Form

The slope-intercept form of the line is (y = mx + b), where (m) is the slope and (b) is the y-intercept. Now that you have the slope, you can use one of the given points to solve for (b).

Step 4: Substitute Known Values into the Slope-Intercept Form

Using the point ((3, 4)) and the slope (m = ⁴⁄₃), you substitute these values into the equation: [4 = \frac{4}{3}(3) + b] Solving for (b): [4 = 4 + b] [b = 0]

Step 5: Interpret the Result

The y-intercept (b) is (0), meaning the line crosses the y-axis at the point ((0, 0)).

Alternative Approach: Using the Two-Point Form

Another way to find the equation of the line (and thus the y-intercept) is by using the two-point form, which is: [y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)] However, this method also leads to finding the slope first and then can be rearranged into the slope-intercept form to find (b), the y-intercept.

Example with Negative Slope

Consider the points ((-2, 3)) and ((4, -1)). The slope (m) is: [m = \frac{-1 - 3}{4 - (-2)} = \frac{-4}{6} = -\frac{2}{3}] Using the point ((-2, 3)): [3 = -\frac{2}{3}(-2) + b] [3 = \frac{4}{3} + b] [b = 3 - \frac{4}{3} = \frac{9}{3} - \frac{4}{3} = \frac{5}{3}] So, the y-intercept is (\frac{5}{3}).

Finding the y-intercept with two points involves a straightforward process of calculating the slope and then using that slope along with one of the points to solve for (b), the y-intercept, in the equation (y = mx + b). This method applies to all lines where two points are known, making it a fundamental tool in geometry and algebra.