Problem 9 - Introductory and Intermediate Algebra for College Students 4th

Question

Write the point-slope form of the equation of the line satisfying each of the conditions in Exercises \(1-28 .\) Then use the point-slope form of the equation to write the slope-intercept form of the equation. Slope \(=-1,\) passing through \(\left(-\frac{1}{2},-2\right)\)

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = -x - \frac{5}{2}\).

Step 1: Plug in values into the point-slope form

Firstly, substitute the given slope \(m = -1\) and point \(\left(-\frac{1}{2}, -2\right)\) (which will be \(x_1\) and \(y_1\)) into the point slope form equation. This means replacing \(m\) with \(-1\), \(x_1\) with \(-\frac{1}{2}\), and \(y_1\) with \(-2\). Doing that, we have: \(y - (-2) = -1 \cdot (x - (-\frac{1}{2}))\).

Step 2: Simplify

Now, simplify the equation. That means dealing with the double negatives, which turns into a positive value. This gives us: \(y + 2 = -1 \cdot (x + \frac{1}{2})\). We can simplify this by distributing the \(-1\) through the parentheses, which gives: \(y + 2 = -x - \frac{1}{2}\).

Step 3: Rewrite in slope-intercept form

The final step is to rearrange this equation into slope-intercept form (\(y = mx + b\)). That requires isolating \(y\) on one side of the equation. To do this, we will subtract \(2\) from both sides of the equation. So the equation simplifies to: \(y = -x - \frac{1}{2} - 2\). Combining like terms gives: \(y = -x - \frac{5}{2}\).

Key Concepts

Point-Slope Form

The point-slope form is a fundamental concept in algebra that represents the equation of a line using a point and the slope. The formula for the point-slope form is \[ y - y_1 = m(x - x_1) \]where:

• \( m \) is the slope of the line,
• \( (x_1, y_1) \) is a given point on the line.

It's a convenient starting point for writing the equation of a line because it directly uses the slope and a specific point to form the equation. This type of formula is very useful when a point and slope are known, but the full equation is not yet needed.
When you substitute values into the point-slope form, the goal is to capture the exact position and direction of a line based on the information provided. For example, if the slope is \(-1\) and the line passes through the point \((-\frac{1}{2}, -2)\), you can quickly place these values into the formula to start constructing the equation.

Slope-Intercept Form

The slope-intercept form of a line is one of the most commonly used forms in algebra. The formula for this form is:\[ y = mx + b \]where:

• \( m \) is the slope of the line, showing the steepness and the direction,
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

This form is especially useful because it quickly reveals the slope and the y-intercept, making it easy to graph the line on the coordinate plane. Transforming an equation from point-slope form to slope-intercept form involves simple algebraic manipulation, such as distributing constants and isolating \( y \).
In our exercise, once the equation in point-slope form was simplified to \( y + 2 = -x - \frac{1}{2} \), switching to slope-intercept form required only a few extra steps to isolate \( y \) by subtracting 2 from both sides, giving us the last output as \( y = -x - \frac{5}{2} \). This shows the crucial aspects of the line, the slope \(-1\) and the y-intercept \(-\frac{5}{2}\).

Equation of a Line

When finding the equation of a line, you have various methods and forms you can use. The equation itself captures the relationship between \( x \) and \( y \) on a coordinate plane, representing a straight line. Knowing how to switch between different forms like point-slope, slope-intercept, and others is a vital skill in algebra.
Each form serves its own purpose and provides different information:

• **Point-slope form**: Useful when you have one point and the slope of the line, letting you start with known values directly.
• **Slope-intercept form**: Works best when you need to quickly graph the line.

In our step-by-step process, we started with a given point and slope to form the point-slope equation. Then, through mathematical simplification, we transitioned to the slope-intercept form. Understanding how to convert between these forms enhances your ability to model real-world situations and improves algebra problem-solving skills.