Problem 18 - A Graphical Approach to College Algebra

Question

Find the slope-intercept form of the equation of the line satisfying the given conditions. Do not use a calculator. Through \((-5,4)\) and \((-3,2)\)

Step-by-Step Solution

Answer

The slope-intercept form is \( y = -x - 1 \).

Step 1: Understanding the Problem

We need to find the equation of a line in slope-intercept form, which is given by the formula \( y = mx + b \). The two points provided are \((-5, 4)\) and \((-3, 2)\). First, let's find the slope \(m\).

Step 2: Calculating the Slope

The formula for the slope \(m\) using two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the points \((-5, 4)\) and \((-3, 2)\) into the formula, we get:\[m = \frac{2 - 4}{-3 + 5} = \frac{-2}{2} = -1\]

Step 3: Finding the Y-intercept

Now that we have the slope \( m = -1 \), we use one of the points to find the y-intercept \(b\). Let's use the point \((-5, 4)\). Plug the slope and the point into the equation:\[ 4 = (-1)(-5) + b \]Solving this, we get:\[ 4 = 5 + b \]\[ b = 4 - 5 = -1 \]

Step 4: Writing the Equation in Slope-Intercept Form

With the slope \( m = -1 \) and the y-intercept \( b = -1 \), the equation of the line in slope-intercept form is:\[ y = -x - 1 \]

Key Concepts

Understanding the Equation of a Line

The equation of a line is a fundamental concept in algebra, representing a straight line in a two-dimensional space. One of the most popular forms to express this equation is the slope-intercept form, which is given by the formula \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept. This form is particularly useful because it provides information about the line's direction and where it crosses the y-axis straightforwardly.

When you have specific points through which the line passes and you're asked to derive its equation in slope-intercept form, these will typically be used to compute the slope and y-intercept. The beauty of this form is its simplicity and directness, making it easy to sketch the graph of the line or relate it to real-world linear scenarios. Understanding each component—slope and y-intercept—is vital to effectively utilizing this equation form.

Mastering Slope Calculation

Calculating the slope of a line is a crucial step in determining the equation in slope-intercept form. The slope \( m \) quantifies the steepness and direction of the line. It is calculated by taking any two points on the line, \((x_1, y_1)\) and \((x_2, y_2)\), and using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula essentially measures the change in \( y \) (rise) divided by the change in \( x \) (run).

It is important to note that:

• A positive slope indicates the line is rising as it moves from left to right.
• A negative slope indicates the line is falling as it moves from left to right.
• A zero slope means the line is horizontal, and an undefined slope corresponds to a vertical line.

In the example provided, calculating the slope using the points \((-5, 4)\) and \((-3, 2)\) yields \( m = -1 \), indicating a line that slopes downward.

Determining the Y-intercept

Once the slope \( m \) is calculated, finding the y-intercept \( b \) is the next step in formulating the equation of a line. The y-intercept is the point where the line crosses the y-axis, which means when \( x = 0 \). It is represented as \( b \) in the slope-intercept form \( y = mx + b \).

To find \( b \), choose one of the points through which the line passes (either \((-5, 4)\) or \((-3, 2)\) in this case) and substitute the values of \( x \), \( y \), and the previously calculated slope \( m \) into the equation. Then solve for \( b \).

Using point \((-5, 4)\), the calculation is as follows:

• Substitute: \( 4 = (-1)(-5) + b \)
• Solve: \( 4 = 5 + b \)
• Result: \( b = -1 \)

Thus, the y-intercept is \( -1 \), revealing that the line crosses the y-axis at this point. With \( m = -1 \) and \( b = -1 \), the complete equation of the line is \( y = -x - 1 \).