Problem 31 - College Algebra and Calculus: An Applied Approach

Question

Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point \(\quad\) Slope \((4,0)\) \(m=-\frac{1}{3}\)

Step-by-Step Solution

Answer

The equation of the line that passes through the point (4,0) and has a slope of -1/3 is \(y = -1/3x + 4/3\).

Step 1: Identify the Given Information

In this problem, a point and the slope of the line are given. The point \((4,0)\) signifies that the x-coordinate is 4 and the y-coordinate is 0, and the slope (\(m\))of the line is given as \(-1/3\).

Step 2: Plug values into slope-intercept formula

Next, plug in the coordinates of the given point and the slope into the slope-intercept equation \(y = mx + c\) to solve for c. Essentially, it will look like this: \(0 = -1/3 * 4 + c\).

Step 3: Solve for c

After plugging in the values into the equation, you can solve for c (which is the y-intercept). This gives the equation as \(0 = -4/3 + c\). Solving for c gives \(c = 4/3\).

Step 4: Write the equation of the line

Finally, you can write-out the equation of the line using the slope and the calculated y-intercept. It is \(y = -1/3x + 4/3\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express the equation of a straight line. This form is incredibly useful because it provides a clear view of both the slope and the y-intercept of the line, which allows us to graph it easily. The formula is as follows:\[ y = mx + c \]Here, \( m \) represents the slope of the line, which indicates the steepness or direction of the line; a positive slope means the line is rising, while a negative slope means it is falling. The \( c \) in the formula is the y-intercept, the point where the line crosses the y-axis.To use the slope-intercept form:

• Identify the slope \( m \).
• Substitute the slope \( m \) and a known point into \( y = mx + c \).
• Solve for \( c \) to find the y-intercept.

Practicing these steps can greatly improve your understanding of linear equations and how they are used to represent lines on a graph.

Coordinate Geometry

Coordinate geometry is a crucial part of mathematics that lets us find geometric information using algebra. This branch of geometry uses a coordinate plane to describe and analyze geometric shapes and their properties, hence its name.In coordinate geometry, a point is defined by an ordered pair \((x, y)\), where \(x\) is the horizontal position, and \(y\) is the vertical position. For example, the point \((4, 0)\) tells us that along the x-axis, our point is at 4, and along the y-axis, it's at 0.Using coordinate geometry:

• We can describe and solve geometric problems analytically.
• We can find distances, midpoints, and slopes using coordinates.
• It allows us to plot lines through given points by using equations like the slope-intercept form.

This type of geometry makes visualizing and solving problems much more manageable, as we could see when finding the y-intercept in the exercise above.

Graphing Lines

Graphing lines involves drawing the representation of a linear equation on a coordinate plane. Understanding how to graph a line helps visualize the relationship between variables and how they change together.To sketch a line, you need at least two pieces of information:

• The slope \( m \) to understand the line's direction and steepness.
• The y-intercept \( c \) to find the starting point on the y-axis.

Once you have these, follow these steps:

• Start by plotting the y-intercept on the y-axis.
• Use the slope to find another point: move up or down (rise) and right or left (run) from the y-intercept.
• Draw a straight line through these points, and extend it in both directions.

The exercise we covered demonstrates this process with the line equation \( y = -\frac{1}{3}x + \frac{4}{3} \). We first plotted \((0, \frac{4}{3})\), used the slope \(-\frac{1}{3}\) to find additional points, and drew the line to represent this equation.