Problem 35 - College Algebra Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((2,4)\) with \(x\) -intercept \(=-2\)

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \( y - 4 = m(x - 2) \) (where \( m \) equals the value calculated in the first step), and in slope-intercept form it is \( y = mx + b \), where \( b \) is calculated by rearranging the equation to solve for \( y \).

Step 1: Find slope of the line using given conditions

First, using the provided x-intercept value and point coordinates, we can calculate the slope of the line. In this case, \( x_1 = -2\), \( y_1 = 0 \) (as the line intercepts at x=-2), and \( x_2 = 2 \), \( y_2 = 4\) are the coordinates of the given point. The formula for the slope (m) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the provided values into the equation to find the slope.

Step 2: Derive the point-slope form of the line by substituting slope and point coordinates into the provided formula

The formula for the point-slope form is:\[ y - y_1 = m(x - x_1) \]Where \( m \) is the slope calculated in the previous step, and \( (x_1, y_1) \) are the coordinates of the given point (2,4). Substitute these values into the formula.

Step 3: Convert the point-slope form to the slope-intercept form

The slope-intercept form is represented by the equation:\[ y = mx + b \]To convert the point-slope form of the line to the slope-intercept form, we isolate y in the equation we derived the previous step.

Key Concepts

Point-Slope Form

The point-slope form is a powerful way to express the equation of a line. It is especially useful when you know the slope of a line and a single point through which the line passes.
The general formula for the point-slope form is given by:

• \[ y - y_1 = m(x - x_1) \]

Here, \( (x_1, y_1) \) are the coordinates of the known point and \( m \) is the slope of the line.
Point-slope form can easily be used to derive other forms of the linear equation, especially when further line characteristics are required.
For example, in the exercise, we have a point \((2,4)\) and an x-intercept at \(-2\). First, we calculate the slope and then use it with the known point to set up the equation in point-slope form.
This helps students see how the positioning of points impacts the line's equation.

Slope-Intercept Form

The slope-intercept form is arguably the most common form of a linear equation, making it a favorite amongst students.

• It is succinct and provides direct insight into two key features of a line: its slope \(m\) and the y-intercept \(b\).
• The equation is structured as:\[ y = mx + b \]

Transforming an equation from the point-slope form to the slope-intercept form primarily involves algebraic manipulation.
By isolating \(y\) in the point-slope formula, one can derive the explicit form where the slope and y-intercept are clearly visible.
This transformation helps when graphing as it immediately tells us where the line crosses the y-axis, which is very intuitive.
In our exercise, after finding the slope using the given intercept and point, substituting appropriately yields the desired slope-intercept form.

Equation of a Line

Understanding the equation of a line allows students to describe any straight line on a graph through algebraic expressions. A line can be represented in several formats, with the most notable ones being point-slope and slope-intercept forms.
Each format offers its advantages depending on what information you are given:

• Point-slope form is useful when a specific point on the line and its slope are known.
• Slope-intercept form is best when the slope and y-intercept should be highlighted.

In practice, understanding these equations is crucial when interpreting or predicting linear behavior in data.
For instance, the line equation derived from our exercise helps visualize its graphical representation and functional relationship between variables.

Slope Calculation

Calculating the slope is the foundational step in expressing linear relationships. The slope describes how steep a line is and is defined as the "rise over run".

• Mathematically, it's calculated using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
• This formula captures the change in \(y\) divided by the change in \(x\) between two points.

For accurate calculations, accurate plotting of points is essential. In the given exercise, we consider two points: the known point \((2,4)\) and x-intercept at \((-2,0)\) to find slope \(m\).
This calculated slope then becomes crucial for writing the line's equation in both point-slope and slope-intercept form.
The slope not only affects the line's angle but also indicates the relationship between the x and y variables.