Problem 74 - Calculus Volume 1

Question

For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form. $$ x \text { -Intercept }=-6 \text { and } y \text { -intercept }=9 $$

Step-by-Step Solution

Answer

The equation of the line is \( y = \frac{3}{2}x + 9 \).

Step 1: Understanding Intercepts

The problem gives us the x-intercept as -6, which means the line crosses the x-axis at (-6, 0). Similarly, the y-intercept is 9, thus the line crosses the y-axis at (0, 9). These intercept points will help us form the equation of the line.

Step 2: Finding the Slope

Using the intercepts, we can find the slope (m) using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, the points are (-6, 0) and (0, 9). Thus, \( m = \frac{9 - 0}{0 + 6} = \frac{9}{6} = \frac{3}{2} \). So, the slope of the line is \( \frac{3}{2} \).

Step 3: Forming the Equation

The slope-intercept form of a line is given by \( y = mx + b \), where m is the slope and b is the y-intercept. With \( m = \frac{3}{2} \) and y-intercept b = 9, the equation becomes \( y = \frac{3}{2}x + 9 \).

Key Concepts

Equation of a Line

The equation of a line is a mathematical expression that gives us valuable information about the line, such as its direction and where it crosses the axes. In mathematics, one of the most common and useful ways to express the equation of a line is in slope-intercept form. The slope-intercept form is written as \( y = mx + b \). Here, \( m \) is the slope of the line, and \( b \) is the y-intercept. This form allows us to quickly understand key features of the line, like how steep it is or where it begins on the y-axis. Unlike other forms like point-slope or standard form, slope-intercept form is especially user-friendly because it immediately communicates what the line looks like and behaves on a graph. For any given line, once we identify its slope and y-intercept, writing its equation becomes straightforward.

X-Intercept

The x-intercept of a line is where the line crosses the x-axis on a graph. At the x-intercept, the value of \( y \) is always zero because the line is not above or below the x-axis at this point; it touches the axis. To find the x-intercept of a line, you can substitute \( y = 0 \) in the equation of the line and solve for \( x \). Knowing the x-intercept helps define the line's position and trajectory. For instance, in our exercise, the x-intercept is -6, meaning the line goes through the point (-6, 0) on the x-axis.

Y-Intercept

The y-intercept is another crucial point in understanding a line's behavior on a graph. This is the point where the line crosses the y-axis, with the x-coordinate being zero. Thus, the y-intercept shows where the line begins from the y-axis perspective. To identify the y-intercept, you substitute \( x = 0 \) into the equation of the line and solve for \( y \). For the equation of a line, the y-intercept is represented by \( b \) in the equation \( y = mx + b \). In this problem, the y-intercept is 9, meaning the line crosses the point (0, 9) on the graph. Selecting this form allows easy visualization and comparison of different lines.

Slope of a Line

The slope of a line, often denoted as \( m \), measures how steep a line is. It tells us how much \( y \) increases or decreases as \( x \) increases by one unit. The slope is crucial for understanding the direction of a line. A positive slope means the line rises as it moves from left to right, while a negative slope implies it falls. To calculate the slope using two points, like the intercepts you might know, we use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For our exercise, using the points (-6, 0) and (0, 9), the slope is \( \frac{9 - 0}{0 + 6} = \frac{3}{2} \). This slope helps us write the line's equation in slope-intercept form, revealing how quickly the line moves upwards due to the positive slope.