Problem 26 - College Algebra with Modeling and Visualization

Question

Find the slope-intercept form for the line satisfying the conditions. y-intercept \(-155,\) slope 5.6

Step-by-Step Solution

Answer

The equation is \( y = 5.6x - 155 \).

Step 1: Identify the Slope-Intercept Form

The slope-intercept form of a linear equation is given by the formula: \[ y = mx + b \]where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Substitute the Slope and Y-intercept

From the given information, we know that the slope \( m = 5.6 \) and the y-intercept \( b = -155 \). Substitute these values into the slope-intercept form equation:\[ y = 5.6x - 155 \]

Step 3: Write the Final Equation

After substituting the known values into the formula, rewrite the final equation of the line in slope-intercept form:\[ y = 5.6x - 155 \]

Key Concepts

Linear Equations

Linear equations are one of the fundamental concepts in algebra. They represent a relationship between variables where the graph of the equation is a straight line. Linear equations have the general form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.
A more commonly used form, particularly for graphing, is the slope-intercept form of a linear equation. This is written as \( y = mx + b \).

• **\( m \)** is the slope of the line.
• **\( b \)** is the y-intercept, which is where the line crosses the y-axis.

Linear equations can be used to model various real-world situations where there is a constant rate of change, such as calculating distance over time or predicting financial growth.

Slope

The slope of a line in a linear equation describes its steepness and direction. Mathematically, it's defined as the ratio of the rise (change in y) over the run (change in x) between two points on the line.
The formula for calculating slope is \( m = \frac{\Delta y}{\Delta x} \). Here, \( \Delta y \) is the difference in y-values, and \( \Delta x \) is the difference in x-values.

• A positive slope means the line inclines upward from left to right.
• A negative slope indicates that the line declines from left to right.
• A zero slope suggests a horizontal line, and an undefined slope indicates a vertical line.

This measure is crucial in fields like physics for calculating speed and acceleration or economics for understanding rates of change in cost or revenue.

Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the term \( b \) represents this intercept.
The y-intercept provides important information.

• It shows the value of \( y \) when \( x = 0 \).
• It can represent starting values in contextual problems, such as initial investments or starting positions.

For example, when given a y-intercept of \(-155\), like in our exercise, it means that the line crosses the y-axis at the point \((0, -155)\). This value is directly substituted into the slope-intercept formula to form the equation of the line, providing a foundation for graphing and further analysis.