Problem 120 - College Algebra with Modeling and Visualization

Question

Explain how you would find the equation of a line passing through two points. Give an example.

Step-by-Step Solution

Answer

The equation of the line is \( y = 2x - 1 \).

Step 1: Identify the Points

First, identify the coordinates of the two points through which the line passes. Let's represent these points as \( (x_1, y_1) \) and \( (x_2, y_2) \). As an example, let's consider the points \( (2, 3) \) and \( (4, 7) \).

Step 2: Calculate the Slope

The slope \( m \) of the line passing through the two points can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the coordinates of the points \( (2, 3) \) and \( (4, 7) \) into this formula: \[ m = \frac{7 - 3}{4 - 2} = \frac{4}{2} = 2 \]

Step 3: Use the Point-Slope Form

The point-slope form of a line is given by:\[ y - y_1 = m(x - x_1) \]Using one of the points, say \( (2, 3) \), and the slope \( m = 2 \), the equation becomes:\[ y - 3 = 2(x - 2) \]

Step 4: Simplify to the Slope-Intercept Form

Simplify the equation from Step 3 to get it in the slope-intercept form \( y = mx + b \): First, distribute the slope: \[ y - 3 = 2x - 4 \]Then, solve for \( y \) by adding 3 to both sides:\[ y = 2x - 4 + 3 \]\[ y = 2x - 1 \] Now, the equation of the line is \( y = 2x - 1 \).

Key Concepts

Slope Formula

To find the equation of a line passing through two points, the first important concept is the slope of the line. The slope is a measure of how steep the line is. To calculate it, we use the **slope formula**. The formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two given points.
The slope tells us how much the y-value of a point on the line changes for each one-unit change in the x-value.
If the slope is positive, the line rises as it moves from left to right. If it's negative, the line falls. A slope of zero means the line is horizontal, and an undefined slope means the line is vertical. **Why is slope important?** Here’s why:

• It provides a consistent way to describe the inclination of the line.
• It is crucial for writing the equation of the line in various forms, such as point-slope and slope-intercept.

The calculation is simple but vital as the foundation for more complex geometry problems.

Point-Slope Form

Once the slope is known, the next step is using it to express the line’s equation efficiently with the **point-slope form**. This form is helpful because it easily incorporates both a point that the line passes through and the slope.
The general formula for point-slope form is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line.
This form allows you to quickly derive the equation when one point and the slope are known. **Highlights of Point-Slope Form:**

• It’s direct and straightforward when you have one point and the slope.
• It can be rearranged to the slope-intercept form, which complements graphical representations.

Thus, it acts as a bridge between initial data (a single point and slope) to the completed equation that is easier to interpret visually.

Slope-Intercept Form

The **slope-intercept form** of a line’s equation is the most popular, especially for graphing or visualizing linear relationships. Once you have the point-slope form \( y - y_1 = m(x - x_1) \), rearranging it into the slope-intercept form \( y = mx + b \) is usually straightforward.
The slope-intercept form is \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept, the point where the line crosses the y-axis.
This form is intuitive and allows you to easily see:

• The rate of change or slope \( m \), describing incline.
• The intercept \( b \), indicating where the line will hit the y-axis.

This format is simple to use in graphing because:

• You draw the intercept on the y-axis, and then apply the slope to find additional points and complete the line.
• It succinctly summarizes both the direction and position of the line.

The slope-intercept form is a practical choice for understanding how changes in \( x \) will influence \( y \) – a skill important in fields like economics, physics, and statistics.