Problem 25 - College Algebra with Modeling and Visualization

Question

Find the slope-intercept form for the line satisfying the conditions. y-intercept \(5,\) slope \(-7.8\)

Step-by-Step Solution

Answer

The slope-intercept form is \( y = -7.8x + 5 \).

Step 1: Identifying Known Values

To write the equation of a line in slope-intercept form \(y = mx + b\), we need the values of the slope \(m\) and the y-intercept \(b\). From the problem, the slope \(m\) is given as \(-7.8\) and the y-intercept \(b\) is \(5\).

Step 2: Substituting Values into the Equation

Substitute the known values into the slope-intercept form equation. Replace \(m\) with \(-7.8\) and \(b\) with \(5\). Thus, the equation becomes: \[ y = -7.8x + 5 \]

Key Concepts

Linear Equations

A linear equation is a powerful tool in mathematics that describes a straight line on a graph. One of the most common forms of expressing a linear equation is the slope-intercept form. In this formulation, the equation is generally expressed as:

• \( y = mx + b \)

where:

• \( y \) represents the dependent variable (often the output or response in real-world scenarios).
• \( x \) is the independent variable (input or cause).
• \( m \) is the slope of the line.
• \( b \) is the y-intercept, where the line crosses the y-axis.

The simplicity of linear equations lies in their ability to predict and model relationships with just two coefficients, \( m \) and \( b \).
They are foundational in understanding more complex mathematical and real-world phenomena because they form the basis for polynomial equations and even calculus. Understanding how to derive and express linear equations can significantly improve problem-solving skills in both mathematical and practical contexts.

Y-Intercept

The y-intercept is a crucial point on the line described by a linear equation. It is the point where the line crosses the y-axis.
In simpler terms, the y-intercept helps us understand where exactly the line begins its journey on a graph concerning the vertical axis. When the linear equation is in the form \( y = mx + b \), the \( b \) value is the y-intercept.
This point always has an x-coordinate of zero because it is where the line meets the y-axis.

• For example, if a line has an equation \( y = -7.8x + 5 \), the y-intercept is \( 5 \).

This means when \( x = 0 \), the value of \( y \) is \( 5 \).
Understanding the y-intercept provides insights into the vertical positioning of a line within a graph. Meaning, no matter where the line is heading (as indicated by the slope), it always starts its path from the y-intercept when \( x \) equals zero.

Slope

The slope of a line is a measure of its steepness and direction. It tells you how much \( y \) changes for a specific change in \( x \).
Symbolized by \( m \) in the slope-intercept form \( y = mx + b \), the slope determines how the line moves across the plane:

• If \( m > 0 \), the line moves upward from left to right.
• If \( m < 0 \), as in our example \( m = -7.8 \), the line slopes downward from left to right.
• A larger absolute value of \( m \) means a steeper line.
• If \( m = 0 \), the line is horizontal, meaning no vertical change as \( x \) changes.

The slope is calculated as the "rise over run," or the change in \( y \) over the change in \( x \). In practical terms, the slope helps in predicting how quickly one variable changes with respect to another.
This is particularly useful in fields like economics and physics, where understanding relationships between different quantities is crucial. By analyzing the slope, one can effectively decipher the rate of change and better interpret the data.