Problem 2 - 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 \((2,4), m=-1\)

Step-by-Step Solution

Answer

The equation is \( y = -x + 6 \).

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We need to write the equation using the given slope and point.

Step 2: Substitute Point and Slope Values

We are given the point \((2, 4)\) and the slope \(m = -1\). Substitute \(x = 2\), \(y = 4\), and \(m = -1\) into the equation \( y = mx + b \). This gives us \( 4 = -1(2) + b \).

Step 3: Solve for the Y-Intercept, b

From the equation \( 4 = -2 + b \), add 2 to both sides to solve for \( b \). This gives \( 4 + 2 = b \) leading to \( b = 6 \).

Step 4: Write the Final Equation

Using the slope \( m = -1 \) and the y-intercept \( b = 6 \), substitute back into the slope-intercept form. The equation of the line is \( y = -x + 6 \).

Key Concepts

Slope

The slope is a fundamental concept when working with linear equations. It measures how steep or flat a line is on a graph. The slope, often represented by the letter \( m \), tells us how much the y-coordinate (vertical position) changes for a change in the x-coordinate (horizontal position). Think of it as the 'rise over run,' or the change in vertical position divided by the change in horizontal position. For example, a positive slope means the line goes upwards as you move from left to right, while a negative slope indicates the line descends. In our exercise, the slope \( m = -1 \) means that for every one unit increase in x, the y value decreases by one unit. This consistent change illustrates the line’s direction and steepness and helps us construct precise line equations.

Y-intercept

The y-intercept is the point where a line crosses the y-axis. This value determines the point at which the x-coordinate is zero. When we say 'y-intercept,' we're referring to the value of \( b \) in the slope-intercept form \( y = mx + b \). It effectively tells us where our line starts on the graph in terms of the y-coordinate when there is no contribution from the x-coordinate. In our example, after calculating the equation using the given point \((2,4)\) and slope \( m = -1 \), we find the y-intercept \( b \) equals 6. This means if you were to graph this line, it would cross the y-axis at the point (0,6). Identifying the y-intercept is crucial because it anchors the line's position on the graph, allowing us to draw the entire line accurately.

Line Equation

The line equation in slope-intercept form is a way to represent a straight line using the formula \( y = mx + b \). Here, \( m \) is the slope, and \( b \) is the y-intercept, both of which define the unique characteristics of the line. This form is particularly useful because it clearly shows the line's slope and starting point, making it easy to graph. In the given exercise, we derived the line equation \( y = -x + 6 \) by substituting the given slope \( m = -1 \) and solving for the intercept using a point on the line \((2,4)\). To graph this equation, you would start at the y-intercept \(6\) on the y-axis, and use the slope \( -1 \) to determine the direction and steepness of the line, moving down one unit for every one unit moved to the right. Understanding how to manipulate and interpret the slope-intercept form of a line is a crucial skill in algebra and serves as a foundation for more advanced mathematical concepts.