Problem 75 - College Algebra

Question

For the following exercises, sketch a line with the given features. Passing through the points (-3,-4) and (3,0)

Step-by-Step Solution

Answer

The line's equation is \( y = \frac{2}{3}x - 2 \).

Step 1: Understand the Problem

We need to sketch a line that passes through two given points: (-3,-4) and (3,0). To start, we will determine the equation of the line because it helps in plotting the line on a graph.

Step 2: Calculate the Slope of the Line

The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the values, \( m = \frac{0 - (-4)}{3 - (-3)} = \frac{4}{6} = \frac{2}{3} \).

Step 3: Use Point-Slope Form to Find the Line Equation

With the slope \( m = \frac{2}{3} \) and a point on the line, say (-3,-4), we use the point-slope form equation: \( y - y_1 = m(x - x_1) \). Substituting, \( y + 4 = \frac{2}{3}(x + 3) \).

Step 4: Simplify to Slope-Intercept Form

Expand the equation from Step 3: \( y + 4 = \frac{2}{3}x + 2 \). Now, solve for \( y \) to get the equation in slope-intercept form: \( y = \frac{2}{3}x - 2 \).

Step 5: Sketch the Line

On a coordinate plane, plot the points (-3,-4) and (3,0). Draw a straight line through these points. The line represents the equation \( y = \frac{2}{3}x - 2 \) and it passes through the given points.

Key Concepts

Slope of a line

The slope of a line is a measure of its steepness, often denoted by the letter \( m \). It can be thought of as the "tilt" of the line when compared to the horizontal axis. To find the slope of a line that passes through two specific points, you use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Let's break this down:

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points through which the line passes.
• The numerator \( (y_2 - y_1) \) indicates how much the line rises or falls as you move from the first point to the second point.
• The denominator \( (x_2 - x_1) \) shows how far you move horizontally between these two points.

In our example, for the points (-3,-4) and (3,0), the slope \( m \) is \( \frac{2}{3} \). This means for every 3 units we move horizontally to the right, the line rises 2 units. The slope is positive, indicating that the line slopes upward as you move from left to right.

Point-slope form

The point-slope form is particularly useful when you have a point on the line and the slope. The formula is:\[ y - y_1 = m(x - x_1) \]In this equation:

• \( m \) is the slope of the line.
• \( (x_1, y_1) \) is a specific point through which the line passes.
• \( x \) and \( y \) are variables representing the coordinates of any further points on the line.

For our line that passes through the point (-3,-4) with a slope \( m = \frac{2}{3} \), the point-slope form would be:\[ y + 4 = \frac{2}{3}(x + 3) \]Here, you adjust the equation based on the known point and the slope to represent that specific line. This form is helpful for quickly writing an equation when you do not yet need it solved for \( y \), which can be useful in some graphing scenarios.

Slope-intercept form

The slope-intercept form of a line equation is one of the most common formats because it directly reveals the slope and the y-intercept. The formula looks like this:\[ y = mx + b \]Where:

• \( m \) stands for the slope of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

To convert an equation into slope-intercept form, solve for \( y \) to make it the subject of the equation. Using the previous step’s point-slope form \( y + 4 = \frac{2}{3}x + 2 \), simplification gives:\[ y = \frac{2}{3}x - 2 \]This indicates that the line crosses the y-axis at \(-2\) with a slope of \( \frac{2}{3} \). This format is especially convenient for graphing lines by hand, as you can quickly identify the line's slope and where it hits the y-axis.