y intercept formula with two points

Understand the Problem

The question is asking for the formula to calculate the y-intercept of a line when two points on that line are given. This typically involves finding the slope using the two points and then applying that to the linear equation format to solve for the y-intercept.

Answer

The formula for the y-intercept is $b = y_1 - \frac{y_2 - y_1}{x_2 - x_1} \cdot x_1$.

Steps to Solve

Identify the two points on the line

Let the two points be $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$. These coordinates are crucial for calculating the slope and then the y-intercept.

Calculate the slope of the line

The slope $m$ is calculated using the formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ This represents the change in $y$ over the change in $x$ between the two points.

Use the slope to set up the linear equation

The equation of the line can be represented in slope-intercept form, which is: $$ y = mx + b $$ Here, $b$ is the y-intercept that we want to find.

Substitute one point into the linear equation

Choose one of the points (let's say $P_1(x_1, y_1)$) and substitute $m$ and the coordinates into the equation: $$ y_1 = mx_1 + b $$

Solve for the y-intercept $b$

Rearranging the equation gives us: $$ b = y_1 - mx_1 $$ Now plug in the values of $y_1$, $m$, and $x_1$ to find $b$.

The formula for calculating the y-intercept $b$ is: $$ b = y_1 - \frac{y_2 - y_1}{x_2 - x_1} \cdot x_1 $$

More Information

The y-intercept is the point where the line crosses the y-axis. It provides important information about the line's behavior and position in the Cartesian plane.

Tips

Failing to correctly substitute values into the slope formula. Always double-check the coordinates you use for $x_1$, $y_1$, $x_2$, and $y_2$.
Confusing the order of points when calculating the slope. Remember to maintain the order as $(x_2, y_2)$ and $(x_1, y_1)$.

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