Problem 6 - College Algebra with Modeling and Visualization

Question

Find a point-slope form of the line satisfying the conditions. Use the first point given for \(\left(x_{1}, y_{1}\right) .\) Then convert the equation to slope-intercept form. Slope 1.7 , passing through \((-8,10)\)

Step-by-Step Solution

Answer

Point-slope form: \(y - 10 = 1.7(x + 8)\); Slope-intercept form: \(y = 1.7x + 23.6\).

Step 1: Understanding Point-Slope Form

The point-slope form of a line is given by the equation \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is a point on the line. In this exercise, the slope \( m \) is 1.7 and the point \( (x_1, y_1) \) is \((-8, 10)\).

Step 2: Substituting Values into Point-Slope Form

Substitute \( m = 1.7 \), \( x_1 = -8 \), and \( y_1 = 10 \) into the point-slope form equation: \( y - 10 = 1.7(x + 8) \).

Step 3: Simplifying the Equation

Distribute the slope on the right side of the equation: \( y - 10 = 1.7x + 13.6 \).

Step 4: Isolating y to Convert to Slope-Intercept Form

Add 10 to both sides to isolate \( y \): \( y = 1.7x + 13.6 + 10 \). Simplify the right side to get \( y = 1.7x + 23.6 \).

Step 5: Final Slope-Intercept Form

The slope-intercept form of the line is \( y = 1.7x + 23.6 \).

Key Concepts

Slope-Intercept Form

The Slope-Intercept form of a linear equation is a very common way to express the equation of a line. It's usually written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept. This form is particularly useful because it directly shows the slope and the y-intercept, allowing you to easily graph the line or understand its behavior.

• Slope \( m \): Indicates the steepness of the line. If \( m \) is positive, the line ascends from left to right; if \( m \) is negative, it descends.
• Y-intercept \( b \): This is the point where the line crosses the y-axis, essentially representing the value of \( y \) when \( x \) equals zero.

To convert from Point-Slope to Slope-Intercept form, such as in the step-by-step solution provided, you would isolate \( y \) on one side of the equation. This reveals the line's slope \( m \) as well as the y-intercept \( b \). The final equation \( y = 1.7x + 23.6 \) is the outcome of this process, where 1.7 is the slope and 23.6 is the y-intercept.

Linear Equations

Linear equations form the foundation of algebra, and they are equations that create a straight line when graphed. The general form of a linear equation in two variables is \( ax + by = c \), where \( a, b, \) and \( c \) are constants. A more common form, especially in graphing, is the slope-intercept form.
Linear equations describe relationships where changes between variables are consistent. This is because they exhibit a constant rate of change, represented by the slope.

• Representation: Commonly, lines are represented by equations such as y = mx + b (slope-intercept) or y - y_1 = m(x - x_1) (point-slope).
• Plotting: You can graph these equations by identifying a point (usually the y-intercept) and applying the slope to plot further points across the coordinate plane.

Understanding how to manipulate and graph linear equations is crucial since it allows us to model and solve many real-world problems, such as calculating costs, estimating growth, or determining distance.

Slope

The slope of a line is a measure of its steepness, often denoted by the letter \( m \). It's calculated as the ratio of the "rise" (change in y) over the "run" (change in x) between two distinct points on the line. Mathematically, it is represented as: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
Slopes can tell us a lot about the direction and angle of a line:

• Positive Slope: The line goes upwards, meaning as \( x \) increases, \( y \) also increases.
• Negative Slope: The line goes downwards, indicating that as \( x \) increases, \( y \) decreases.
• Zero Slope: A perfectly horizontal line, indicating no change in \( y \) as \( x \) changes.
• Undefined Slope: A vertical line, where \( x \) remains constant regardless of changes in \( y \).

In the provided step-by-step solution, the slope is given as 1.7. This positive slope means that with each unit increase in \( x \), \( y \) increases by 1.7 units, producing a line that rises from left to right. Understanding slope is essential for interpreting how variables in a line are related to one another.