Finding the Y-intercept from two points is an essential skill in mathematics. The Y-intercept represents the point at which a curve or line intersects the Y-axis on a graph. By discovering this value, we can obtain valuable information about the equation of the line or curve we are working with. This article will guide you through the process of finding the Y-intercept using two given points.

Let’s assume we are given two points: (x₁, y₁) and (x₂, y₂). Our objective is to determine the Y-intercept, denoted as “b,” using these two points.

To begin, we need to identify the slope of the line passing through these two points. The slope, denoted by “m,” can be determined using the slope formula – which states that the slope (m) equals the change in Y divided by the change in X.

m = (y₂ – y₁) / (x₂ – x₁)

By substituting the given coordinates into the formula, we can calculate the slope.

Once we have found the slope, we can proceed to use one of the two points to derive the equation of the line using the slope-intercept form: y = mx + b. Here, “m” represents the slope, and “b” represents the Y-intercept.

To find the equation, we can select either of the two points, let’s say (x₁, y₁). By substituting its coordinates and the value of the slope into the slope-intercept form, we can solve for the Y-intercept, “b.”

y = mx + b
y₁ = m * x₁ + b

We can then rearrange this equation to isolate “b”:

b = y₁ – m * x₁

The value we obtain for “b” is the numerical representation of the Y-intercept.

For example, let’s say we are given the points (3, 5) and (7, 9). We can find the slope by calculating:

m = (9 – 5) / (7 – 3)
m = 4 / 4
m = 1

Now that we have the slope, we can choose one of the points, let’s say (3, 5), and substitute the values into the slope-intercept form:

5 = 1 * 3 + b

By simplifying this equation, we can isolate “b”:

b = 5 – 3
b = 2

Therefore, the Y-intercept (b) of the line passing through the points (3, 5) and (7, 9) is 2.

Finding the Y-intercept from two points enables us to understand how a line intersects the Y-axis and to formulate the equation of that line. This knowledge can be useful in various mathematical and practical scenarios, such as graphing functions, determining linear regression models, or analyzing data trends.

In conclusion, determining the Y-intercept from two points involves finding the slope using the slope formula and then using one of the points to derive the equation of the line using the slope-intercept form. By solving for “b,” we can find the Y-intercept and gain valuable insights into the equation and behavior of the line or curve.

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