Problem 50 - Precalculus with Limits

Question

In Exercises 41-50, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.) \((-1, -6)\), \(m=-\frac{1}{2}\)

Step-by-Step Solution

Answer

The additional three points through which the line passes are (0,-6.5), (1,-7), (-2,-6).

Step 1: Express the Equation of a Line

Start by plugging in \(m=-\frac{1}{2}\) and the point \((-1, -6)\) into the slope-intercept form of the line. Rearrange to find the y-intercept (b). The equation becomes: -6= -\frac{1}{2}(-1) + b, which simplifies to -6= \frac{1}{2} + b.

Step 2: Solve for b

Subtract \(\frac{1}{2}\) from both sides of the equation to isolate the b term. The calculation becomes \(-6 - \frac{1}{2} = b\), which simplifies to \(-\frac{13}{2} = b\). So, the equation of the line is \(y = -\frac{1}{2}x - \frac{13}{2}\).

Step 3: Test Different x-values to get the points

Now, substitute different x values into the line equation to get the corresponding y values. Let's choose 0, 1, and -2 for the x values. For x=0: \(y = -\frac{1}{2}(0) - \frac{13}{2}\) so the corresponding point is (0,-6.5), for x=1: \(y = -\frac{1}{2}(1) - \frac{13}{2}\), so the corresponding point is (1,-7), and for x=-2: \(y = -\frac{1}{2}(-2) - \frac{13}{2}\), so the corresponding point is (-2,-6).

Key Concepts

Linear Equations

A linear equation is a type of equation that describes a straight line on a graph. It is commonly represented in the slope-intercept form as \( y = mx + b \).
This equation is made up of two main components: the slope (\( m \)) and the y-intercept (\( b \)). The slope \( m \) tells you how steep the line is, and if it is positive, negative, or flat. The y-intercept \( b \) defines where the line crosses the y-axis.

Understanding linear equations is crucial because they are used to model relationships between variables in a variety of fields, such as physics, economics, and social sciences. They help predict outcomes and understand trends based on given data.

Finding Points

Finding points on a linear equation involves substituting different x-values into the equation to get corresponding y-values. This helps in plotting the graph of the equation.

To find points, use the equation in the slope-intercept form. For instance, in the equation \( y = -\frac{1}{2}x - \frac{13}{2} \):

• Choose any x-value (like 0, 1, or -2).
• Substitute the chosen x-value into the equation.
• Solve for the y-value to get a coordinate pair \((x, y)\).

This process will give you points through which the line passes.
Having multiple points is useful to confirm that your graph is accurate and to understand the behavior of the line across the graph.

Slope

The slope of a line represents how much y changes for a unit change in x. It is often described as "rise over run," meaning how much the line rises or falls vertically as it moves horizontally.

The formula for slope \( m \) is \( m = \frac{\text{Change in } y}{\text{Change in } x} \). A positive slope indicates that the line increases, moving from left to right. Conversely, a negative slope indicates that the line decreases over that direction. In our example, the slope \( m = -\frac{1}{2} \) tells us that for every 2 units the line moves to the right on the x-axis, it moves 1 unit down on the y-axis.

Understanding slope helps in predicting how changes in one variable affect another in models that use linear equations.