We want to write an equation for the line that passes through points $(0, b)$ and $(x, y)$.

Let's start with the equation for the slope of a line, $m$, through any two points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$.
$\frac{y_2-y_1}{x_2-x_1}=m$

Step 1: Complete the substitution, using $(0, b)$ for $\left(x_1, y_1\right)$ and $(x, y)$ for $\left(x_2, y_2\right)$ in the above equation.
$\frac{y-\square}{\square-\square}=m$

Step 2: Simpl

- Substitute the given points \((0, b)\) and \((x, y)\) into the slope formula to get \(\frac{y-b}{x-0}=m\). - Simplify the denominator to obtain \(\frac{y-b}{x}=m\). - Eliminate the fraction by multiplying both sides by \(x\), resulting in \(y-b=m \cdot x\). - Solve for \(y\) by adding \(b\) to both sides, which yields the slope-intercept form: \(\boxed{y=m \cdot x+b}\). ### Explanation 1. Understanding the Problem and Initial Setup We are given the general formula for the slope of a line, \(m = \frac{y_2 - y_1}{x_2 - x_1}\), and two specific points that the line passes through: \((x_1, y_1) = (0, b)\) and \((x_2, y_2) = (x, y)\). Our goal is to use these points and the slope formula to derive the slope-intercept form of a linear equation, which is \(y = mx + b\). We will complete the missing parts in the provided steps. 2. Step 1: Substituting the Points into the Slope Formula The first step is to substitute the coordinates of the given points, \((0, b)\) and \((x, y)\), into the slope formula. We replace \(y_1\) with \(b\), \(x_1\) with \(0\), \(y_2\) with \(y\), and \(x_2\) with \(x\). This gives us: \[\frac{y-b}{x-0}=m\] 3. Step 2: Simplifying the Denominator Next, we simplify the denominator of the expression. Since \(x - 0 = x\), the equation becomes: \[\frac{y-b}{x}=m\] 4. Step 3: Eliminating the Fraction To eliminate the fraction, we multiply both sides of the equation by \(x\). This isolates the term \(y-b\) on the left side: \[y-b=m \cdot x\] 5. Step 4: Solving for y Finally, to solve for \(y\) and arrive at the slope-intercept form, we add \(b\) to both sides of the equation: \[y=m \cdot x+b\] This is the well-known slope-intercept form of a linear equation, where \(m\) is the slope and \(b\) is the y-intercept. ### Examples Understanding how to derive the slope-intercept form of a line is fundamental in many real-world applications. For instance, if you are tracking the growth of a plant over time, you can plot its height at different intervals. If the growth is consistent, you can use two data points to find the equation of the line representing its growth. This equation can then predict the plant's height at any given time or determine its initial height (y-intercept) and growth rate (slope). This concept is also crucial in fields like physics for analyzing motion, in economics for modeling supply and demand, and in engineering for designing systems with linear relationships.