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Chapter 5: Problem 30

Sketch the line with the given slope and \(y\) -intercept. $$m=\frac{2}{3},(0,-1)$$

Short Answer

Expert verified
The line has the equation \( y = \frac{2}{3}x - 1 \). Plot (0, -1), move right 3 and up 2 to plot (3, 1), and draw the line through these points.

Step by step solution

01

Identify the equation of the line

To sketch a line when given the slope and y-intercept, first identify the equation of the line in the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

Substitute the given values

Substitute the given slope \( m = \frac{2}{3} \) and y-intercept \( b = -1 \) into the equation. The equation becomes \( y = \frac{2}{3}x - 1 \).
03

Plot the y-intercept

Start by plotting the y-intercept on the Cartesian plane. The y-intercept is \( (0, -1) \). Place a point at this location.
04

Use the slope to find another point

The slope \( \frac{2}{3} \) means that for every 3 units you move to the right, move up 2 units. From the point (0, -1), move 3 units to the right to (3, -1) and then up 2 units to the point (3, 1).
05

Draw the line

Using a ruler, draw a straight line through the points (0, -1) and (3, 1). Extend the line across the grid, adding arrows on both ends to indicate that it continues indefinitely.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
The slope-intercept form is a straightforward way to describe a straight line using an equation. It is written as \( y = mx + b \). In this formula, \( m \) represents the slope of the line and \( b \) is the y-intercept. The y-intercept is the point where the line crosses the y-axis.
  • Slope (\( m \)): Indicates the steepness of a line. It is calculated as "rise over run" or the change in vertical direction divided by the change in the horizontal direction.
  • Y-Intercept (\( b \)): The point where the line crosses the y-axis, shown as \((0, b)\). It gives information about the line's position with respect to the y-axis.
Understanding the slope-intercept form is crucial because it simplifies the process of graphing a line, especially when both the slope and y-intercept are known. This formulation makes it easier to compute and interpret geometric properties of the line.
Graphing a Line
Graphing a line, particularly using the slope-intercept form, involves plotting the y-intercept first and then using the slope to determine another point on the line. Here鈥檚 a simple step-by-step to accomplish this:
  • Start with the y-intercept: Locate the y-intercept on the graph. For instance, if the y-intercept is \(-1\), plot this on the y-axis at the point \((0, -1)\).
  • Apply the slope: Use the slope to find a second point. A slope of \( \frac{2}{3} \) implies from the y-intercept move 3 units to the right (horizontal change) and 2 units up (vertical change) to find the next point.
  • Draw the line: Connect the y-intercept and the second point using a straight edge to form a continuous line extending in both directions. Add arrows at each end to signify that the line extends infinitely.
This method of graphing ensures accuracy and utilizes both the slope and y-intercept effectively, offering a direct visual representation of the equation.
Equation of a Line
The equation of a line is essentially the mathematical representation of a straight line on a plane. The slope-intercept form, \( y = mx + b \), is a popular format due to its clarity and efficiency in line representation. Here are some insights into why it is paramount:
  • Linear Relationships: This equation accurately describes linear relationships, often seen in real-world data. It's a foundation for understanding more complex algebraic concepts.
  • Simplicity and Clarity: By directly providing the slope and y-intercept, this form makes it simple for anyone to visualize the line's behavior and position almost immediately.
  • Versatility: Once mastered, this form allows for quick conversions into other forms, such as point-slope or standard form, giving flexibility in different mathematical scenarios.
In summary, mastering the equation of a line enables individuals to efficiently interpret and manipulate linear relationships, both in mathematical exercises and real-world applications.

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