Problem 28 - Precalculus Mathematics for Calculus

Question

Find an equation of the line that satisfies the given conditions. \(x\) -intercept \(-8 ; \quad y\) -intercept 6

Step-by-Step Solution

Answer

The equation of the line is \(y = \frac{3}{4}x + 6\).

Step 1: Understanding Intercepts

The x-intercept is where the line meets the x-axis, meaning the value of y at this point is zero. Similarly, the y-intercept is where the line meets the y-axis, meaning the value of x at this point is zero.

Step 2: Identifying the x-intercept

We are given the x-intercept as -8. This means the point (-8, 0) is on the line.

Step 3: Identifying the y-intercept

We are given the y-intercept as 6. This means the point (0, 6) is on the line.

Step 4: Finding the Slope of the Line

The slope (m) of the line can be found using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Using the points (-8, 0) and (0, 6), we calculate: \(m = \frac{6 - 0}{0 - (-8)} = \frac{6}{8} = \frac{3}{4}\).

Step 5: Writing the Equation in Point-Slope Form

Using the slope found and one of the intercept points, you can use the point-slope form equation \(y - y_1 = m(x - x_1)\). Using point (-8, 0): \(y - 0 = \frac{3}{4}(x + 8)\).

Step 6: Converting to Slope-Intercept Form

Simplifying the point-slope equation to slope-intercept form \(y = mx + b\):\[y = \frac{3}{4}x + \frac{3}{4} \times 8\]\[y = \frac{3}{4}x + 6\].So, the equation of the line is \(y = \frac{3}{4}x + 6\).

Key Concepts

Understanding the x-intercept

The x-intercept of a line is the point where it crosses the x-axis. At this point, the value of y is zero because the line hasn't moved upwards or downwards on the Cartesian plane yet. In our exercise, we are given the x-intercept as -8. This simply means that when y is equal to zero, the x value is -8. Therefore, the coordinates of the x-intercept are (-8, 0). This stepping stone is essential for building the equation of the line as it gives us a precise point through which the line passes.

Decoding the y-intercept

Conversely, the y-intercept is the point at which the line crosses the y-axis. At this point, the value of x is zero because the line hasn't moved to the left or right yet. For our task, we've been provided with a y-intercept of 6. This indicates that when x equals zero, y has a value of 6. Thus, the coordinates representing the y-intercept in this exercise are (0, 6). Knowing the y-intercept is critical for expressing the line equation in the most commonly used form, which we'll explore later on.

Performing Slope Calculation

To find the slope or the steepness of the line, the formula used is:

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

In our case, the two points given are (-8, 0) and (0, 6). The calculation proceeds as follows:
\( m = \frac{6 - 0}{0 - (-8)} = \frac{6}{8} = \frac{3}{4} \).This value, \(\frac{3}{4}\), tells us how much y increases for every one-unit increase in x. It provides the 'tilt' or direction of the line, showing the careful balance between its vertical and horizontal shifts.

Understanding the Point-Slope Form

Point-slope form is a compelling way to express the equation of a line, especially using a known point and the slope. It is structured as:

• \( y - y_1 = m(x - x_1) \)

Substituting the point (-8, 0) and the slope \(\frac{3}{4}\), we get:
\( y - 0 = \frac{3}{4}(x + 8) \).Using this form simplifies the process of finding the equation of a line when a point and slope are known, bridging us to the more familiar slope-intercept form.

The Slope-Intercept Form Revealed

Slope-intercept form is perhaps the most widely recognized form to articulate a line equation. It is denoted as:

• \( y = mx + b \)

Here, \( m \) represents the slope, and \( b \) is the y-intercept. By converting our point-slope form equation:
\( y = \frac{3}{4}x + \frac{3}{4} \times 8 \),we simplify it to:
\( y = \frac{3}{4}x + 6 \).This explicit equation \( y = \frac{3}{4}x + 6 \) now effortlessly communicates the line's slope and where it crosses the y-axis, making graphing straightforward and intuitive.