Problem 16 - 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. Passing through \((3,5)\) and \((8,15)\)

Step-by-Step Solution

Answer

The slope-intercept form of the equation of a line passing through \((3,5)\) and \((8,15)\) is \(y = 2x - 1\).

Step 1: Determine the Slope

The equation to find the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting in our given points (3,5) and (8,15), we compute the slope as \(m = \frac{15-5}{8-3} = \frac{10}{5} = 2\).

Step 2: Write the Point-Slope Form

The point-slope form of a line is \(y - y_1 = m(x - x_1)\). Using one of our points, say \((3,5)\) and the calculated slope of \(2\), the equation of the line in point-slope form becomes \(y - 5 = 2(x - 3)\).

Step 3: Convert to Slope-Intercept Form

The slope-intercept form of a line is \( y=mx+b \). To convert to this form, we need to isolate \( y \) in our point-slope equation. This results in: \(y = 2x - 6 + 5\), so the final slope-intercept form is \(y = 2x - 1\).

Key Concepts

Slope Calculation

Calculating the slope of a line is an essential skill in algebra that tells us how steep the line is. The slope, often represented by \( m \), measures the change in the vertical position (the rise) for a unit change in the horizontal position (the run) between two points. To find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the formula:

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

Substituting in the points \((3, 5)\) and \((8, 15)\), we calculate:

• Rise = \( 15 - 5 = 10 \)
• Run = \( 8 - 3 = 5 \)
• Slope \( m = \frac{10}{5} = 2 \)

The result \( m = 2 \) tells us that for each step to the right along the x-axis, the line goes 2 steps up.

Slope-Intercept Form

The slope-intercept form of a line is one of the most common ways to express the equation of a line. It emphasizes the slope and the y-intercept, which is where the line crosses the y-axis. The formula for the slope-intercept form is:

• \( y = mx + b \)

Here, \( m \) represents the slope, and \( b \) represents the y-intercept. Knowing these two values allows us to understand and sketch the line quickly on a graph.
For our problem, we must transform the point-slope form equation we derived into the slope-intercept form. Beginning with the equation \( y - 5 = 2(x - 3) \), we simplify it:

• First, expand: \( y - 5 = 2x - 6 \)
• Then, add 5 to both sides to solve for \( y \):
  \( y = 2x - 6 + 5 \)
• This simplifies to \( y = 2x - 1 \)

In the final equation \( y = 2x - 1 \), 2 is the slope and -1 is the y-intercept. This shows the line crosses the y-axis at \((0, -1)\).

Equation of a Line

An equation of a line provides a complete description of the line’s path across a graph. There are different forms, but the two most commonly used are the point-slope form and the slope-intercept form.
The point-slope form is particularly useful when you know a point on the line and the slope. The formula is:

• \( y - y_1 = m(x - x_1) \)

Using a known point \((3, 5)\) and our calculated slope \(2\), we get: \( y - 5 = 2(x - 3) \). This form tells us the slope and a specific point the line passes through, giving us a precise way to sketch a line.
On the other hand, the slope-intercept form \( y = mx + b \) puts emphasis on the overall slope of the line and where it crosses the y-axis. Both forms give critical insights into the line's behavior and how it fits within a graph.