Problem 6 - A Graphical Approach to College Algebra

Question

Write the slope-intercept form of the line that passes through the given point with slope \(m .\) Do not use a calculator. Through \((-5,9), m=-0.75\)

Step-by-Step Solution

Answer

The equation is \( y = -0.75x + 5.25 \).

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a linear equation is \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept. Our goal is to find the value of \( b \) that fits the given line.

Step 2: Substitute the Known Values

We are given the slope \( m = -0.75 \), and a point \((-5, 9)\). Substitute \( m \), \( x = -5 \), and \( y = 9 \) into the equation to find \( b \): \[ 9 = -0.75(-5) + b \]

Step 3: Solve for b

Calculate \(-0.75 \times -5\):\[ -0.75 \times -5 = 3.75 \]Substitute back into the equation: \[ 9 = 3.75 + b \]Now solve for \( b \):\[ b = 9 - 3.75 = 5.25 \]

Step 4: Write the Final Equation

Now that we have found \( b = 5.25 \), substitute \( m \) and \( b \) back into the slope-intercept form:\[ y = -0.75x + 5.25 \] This is the equation of the line in slope-intercept form.

Key Concepts

Linear Equation

A linear equation is a type of equation that creates a straight line when it is graphed on a coordinate plane. This form of equation is crucial for understanding relationships that increase or decrease at a constant rate.
In general, a linear equation is expressed in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. However, the most common form we use, especially when discussing slope-intercept form, is \(y = mx + b\).

• The term "linear" signifies that the equation represents a line.
• The equation shows a direct relationship between two variables, \(x\) and \(y\).
• The solution set of a linear equation is all the points \((x, y)\) that lie on the line.

This means that every time x increases or decreases, y changes in a predictable linear fashion. Understanding this lays the foundation for topics like algebra and calculus.

Slope

Slope is a measure of how steep a line is. It is often described as the rise over the run, or the change in \(y\) divided by the change in \(x\).
In the context of the slope-intercept form \(y = mx + b\), \(m\) represents the slope.

• A positive slope means the line ascends as you move from left to right.
• A negative slope means the line descends as you move from left to right.
• A slope of zero indicates a horizontal line, where there is no change in \(y\) as \(x\) changes.

The given slope in the original exercise was \(-0.75\), indicating a gentle descent. Understanding slope is crucial as it defines the rate of change between the two variables represented by the line.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. It is a fundamental part of the slope-intercept form equation \(y = mx + b\). Here, \(b\) represents the y-intercept.

• The y-intercept is the value of \(y\) when \(x = 0\).
• It tells us where the line begins on the y-axis before it starts to ascend or descend based on the slope.

In the exercise, once the equation was solved, the y-intercept \(b\) was found to be \(5.25\). This means that when \(x\) is zero, the value of \(y\) will be \(5.25\). Understanding the y-intercept helps us locate the line on the coordinate plane, giving us crucial information about the starting point of the line.