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Chapter 1: Problem 20

In Exercises 17-28, find the slope and \(y\)-intercept (if possible) of the equation of the line. Sketch the line. \( y = -\frac{3}{2}x + 6 \)

Short Answer

Expert verified
The slope of the line is -\frac{3}{2} and the y-intercept is 6.

Step by step solution

01

Identify the Slope

Looking at the given linear equation, we can see it is already in slope-intercept form. The coefficient of x is the slope of the line. Here, the slope (m) is -\frac{3}{2}.
02

Identify the y-intercept

The y-intercept (b) is the constant term in the equation. Here, the y-intercept is 6.
03

Sketch the Line Using Slope and y-intercept

Start by plotting the y-intercept (0,6) on the graph. From there, use the slope to determine where the next point on the line will be. The slope is the ratio of the vertical change (rise) over the horizontal change (run). A negative slope means the line is decreasing, so from the point (0,6), move down 3 units and right 2 units (because slope is -\frac{3}{2}) to find the next point and connect the points to create the line.

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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 way to easily write the equation of a line. It is typically expressed as \( y = mx + b \). This format reveals two essential features of the line:
  • \( m \), which represents the slope of the line.
  • \( b \), which is the y-intercept, or the point where the line crosses the y-axis.
A slope-intercept form is particularly useful because it provides immediate insight into the slope and placement of the line on a graph. These two components 鈥� slope and y-intercept 鈥� allow us to quickly graph the equation or understand the line's direction and position.
Graphing Linear Equations
Graphing a linear equation using the slope-intercept form is straightforward. Here's a step-by-step method:
  • Start by identifying the y-intercept \( b \), which is the point where the line crosses the y-axis. In the equation \( y = -\frac{3}{2}x + 6 \), the y-intercept is 6, so you plot the point \((0, 6)\) on the graph.
  • Next, use the slope \( m \) to find another point. Slope \( m = -\frac{3}{2} \) means that for every 2 units you move to the right (run), you move down 3 units (rise). From the point \((0, 6)\), move right 2 units and down 3 units to locate the next point.
  • Draw a line through the points. Extend it in both directions to cover the graph area. This line is the graphical representation of the equation.
By understanding these steps, you can efficiently graph any line given in slope-intercept form.
Slope and Y-Intercept
The slope and y-intercept are critical components that define the behavior of a line:The **slope** \( m \) indicates how steep the line is, and its sign (positive/negative) reveals the direction:
  • A positive slope means the line ascends as it moves from left to right.
  • A negative slope, like \(-\frac{3}{2}\), signifies a descent.
  • The absolute value of the slope shows the degree of steepness or how quickly the line rises or falls.
The **y-intercept** \( b \) is the starting point of the line on the y-axis:
  • It's the value of \( y \) when \( x \) is 0, making it easy to plot the initial starting point for drawing your line.
  • This feature simplifies the equation's interpretation and is crucial for quickly sketching a graph.
Together, understanding slope and y-intercept translates the equation of a line into a visual slope, aiding in comprehension and graphing.

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