Problem 23 - Introductory Algebra for College Students

Question

Write the point-slope form of the equation of the line satisfying each of the conditions in Exercises. Then use the point-slope form of the equation to write the slope-intercept form of the equation. Passing through \((-3,-1)\) and \((4,-1)\)

Step-by-Step Solution

Answer

The slope-intercept form of the equation is \(y = -1\).

Step 1: Calculate the slope

We need to calculate the slope using the two given points. The formula for the slope is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting \((-3,-1)\) into \((x_1, y_1)\) and \((4,-1)\) into \((x_2, y_2)\) we get \(m = \frac{-1 -(-1)}{4 - (-3)} = 0\)

Step 2: Write the point-slope form

The point-slope form for a line is \(y - y_1 = m(x - x_1)\). From Step 1 we know m=0 and one of the points is (-3, -1). So the equation would be \(y - (-1) = 0 (x - (-3))\). This simplifies to \(y + 1 = 0\) or \(y = -1\)

Step 3: Convert to slope-intercept form

The slope-intercept form of an equation is \(y = mx + b\). Here, the calculated slope is 'm = 0' from step 1 and 'b' is the y-coordinate from step 2 which is \(y = -1\). So, the equation becomes \(y = 0x -1\), which further simplifies to \(y = -1\).

Key Concepts

Slope Calculation

Calculating the slope of a line is essential when working with linear equations. The slope tells us how steep a line is and the direction it has. The standard formula for slope between two points, \((x_1, y_1)\) and \((x_2, y_2)\), is:

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

In the example, the points \((-3, -1)\) and \((4, -1)\) are given. Substituting these into the slope formula, we find:

• \( m = \frac{-1 - (-1)}{4 - (-3)} = \frac{0}{7} = 0 \)

The slope of 0 indicates a horizontal line. This means there is no vertical change as you move along the line. Understanding that a zero slope represents a horizontal line is very useful in recognizing and describing the behavior of lines.

Slope-Intercept Form

The slope-intercept form is a concise way to express linear equations. Given by:

• \( y = mx + b \)

Here, \( m \) is the slope, and \( b \) is the y-intercept, where the line crosses the y-axis. From the previous calculations, we identified the slope \( m = 0 \) and found that our equation simplifies to \( y = -1 \).
This simplified equation aligns with the slope-intercept form, showing that the line is horizontal and passes through the y-axis at \( y = -1 \). Thus, the slope-intercept form, in this case, is straightforward and visually represents what happens on the graph.

Line Equations

Understanding line equations involves using different forms to express the equation of a line. Two common forms are:

• Point-Slope Form: \( y - y_1 = m(x - x_1) \)
• Slope-Intercept Form: \( y = mx + b \)

In the exercise, we started with the point-slope form using one of the points

• \((-3, -1)\) and slope \( m = 0 \)

Plugging these into the point-slope form results in:

• \( y - (-1) = 0(x - (-3)) \)
• Which simplifies to: \( y = -1 \)

Converting between these forms helps in identifying the characteristics of a line quickly. Each form offers different insights, such as pointing out the direction and intercepts, enriching your understanding of linear relationships.