Problem 21 - 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 \((-4,-1)\) and \((3,4)\)

Step-by-Step Solution

Answer

The equation of the line passing through the points (-4,-1) and (3,4) in slope-intercept form is \(y = \frac{5}{7}x - 1\).

Step 1: Calculate the Slope

First, find the slope of the line using the formula \(m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\). In this case, \(x_{1} = -4, y_{1} = -1, x_{2} = 3\), and \(y_{2} = 4\). Plugging these values into the formula gives \(m = \frac{4 - (-1)}{3 - (-4)} = \frac{5}{7}\). So, the slope of the line is \(\frac{5}{7}\).

Step 2: Insert values into Point-Slope Form

Next, substitute the values for \(m\), \(x_{1}\), and \(y_{1}\) into the point-slope form equation: \(y - y_{1} = m(x - x_{1})\). This gives \(y + 1 = \frac{5}{7}(x + 4)\).

Step 3: Convert into Slope-Intercept Form

The final task is to rearrange the equation from step 2 into the slope-intercept form \(y = mx + b\). To do so, distribute \(\frac{5}{7}\) into \(x + 4\) getting \(y + 1 = \frac{5}{7}x + \frac{20}{7}\). Subtracting 1 from both sides gives the final equation, \(y = \frac{5}{7}x - \frac{7}{7}\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most popular ways to express the equation of a line. It's written as \(y = mx + b\), where:

• \(m\) is the slope of the line
• \(b\) is the y-intercept, which is where the line crosses the y-axis

This form is useful because it allows you to quickly determine both the slope of the line and the y-intercept just by looking at the equation.

When you're given the point-slope form of a line, like in the original exercise, you start by distributing the slope across the terms in parentheses. In our example, this looked like \(y + 1 = \frac{5}{7}(x + 4)\). After distributing \(\frac{5}{7}\) to each part inside the parentheses, you simplify to get to the form \(y = \frac{5}{7}x - \frac{7}{7}\). This is the format you want for slope-intercept form, making it easy to see the slope and intercept right away.

Equation of a Line

The equation of a line serves as a mathematical representation of the line in a coordinate plane. There are various forms to write the equation, each suited for different applications or given conditions. In mathematics, lines can be described using:

• Point-Slope Form: Here's where you use a single point on the line and the slope. It's written as \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a known point on the line.
• Slope-Intercept Form: As mentioned before, it appears as \(y = mx + b\).

To derive an equation of a line, information such as a point through which the line passes and the slope is necessary. In the original exercise, you were given two points and used them to calculate the slope, then inserted one of those points into the point-slope form.

Slope Calculation

Slope is a measure of how steep a line is. It's calculated as the change in the y-coordinates divided by the change in the x-coordinates, or simply rise over run. The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

In the original exercise, for point \((-4, -1)\) and \((3, 4)\), you calculate the slope by substituting these points into the formula. This gives \(m = \frac{4 - (-1)}{3 - (-4)} = \frac{5}{7}\), indicating the line rises 5 units for each 7 units it moves horizontally.

• A positive slope means the line is going upward from left to right.
• A negative slope means the line is going downward from left to right.
• Zero slope means the line is horizontal, while an undefined slope is for vertical lines.

Understanding the slope helps not just in geometric intuition, but in predicting how changes in x affect y in linear equations.