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Chapter 10: Problem 4

A line has equation \(y=-1.5 x+1\). a. Pick five distinct \(x\) -values, use the equation to compute the corresponding \(y\) -values, and plot the five points obtained. b. Give the value of the slope of the line; give the value of the \(y\) -intercept.

Short Answer

Expert verified
The slope is \(-1.5\), and the \(y\)-intercept is \(1\).

Step by step solution

01

Identify the Linear Equation components

Given the equation \( y = -1.5x + 1 \), recognize that it is in the slope-intercept form \( y = mx + b \), where \( m \) represents the slope and \( b \) the \( y \)-intercept.
02

Determine the Slope and Y-Intercept

From the equation \( y = -1.5x + 1 \), identify the slope \( m = -1.5 \) and the \( y \)-intercept \( b = 1 \).
03

Select X-values

Choose five distinct \( x \)-values for computing \( y \)-values. Let's pick \( x = -2, -1, 0, 1, 2 \).
04

Compute Y-values

For each selected \( x \)-value, insert into the equation \( y = -1.5x + 1 \) to find the \( y \)-value. * \( x = -2 \): \( y = -1.5(-2) + 1 = 3 + 1 = 4 \)* \( x = -1 \): \( y = -1.5(-1) + 1 = 1.5 + 1 = 2.5 \)* \( x = 0 \): \( y = -1.5(0) + 1 = 0 + 1 = 1 \)* \( x = 1 \): \( y = -1.5(1) + 1 = -1.5 + 1 = -0.5 \)* \( x = 2 \): \( y = -1.5(2) + 1 = -3 + 1 = -2 \)
05

Plot the Points

Plot the points \( (-2, 4) \), \( (-1, 2.5) \), \( (0, 1) \), \( (1, -0.5) \), and \( (2, -2) \) on a graph and draw the line passing through them. Observe that they form a straight line consistent with the slope and intercept derived.

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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 of a linear equation is one of the most straightforward ways to express the equation of a line. In this form, the equation is written as \( y = mx + b \). Here, \( m \) stands for the "slope" of the line, and \( b \) is the "y-intercept," which is the point where the line crosses the y-axis. Understanding this form is crucial because it easily reveals key characteristics of the line:
  • Slope \( (m) \): This indicates the steepness or inclination of the line. If the slope is positive, the line ascends as it moves from left to right. Conversely, a negative slope means the line descends in that direction.
  • Y-intercept \( (b) \): This tells us where the line intersects the y-axis. At this point, \( x \) is always equal to zero.
These components help in quickly sketching graphs, understanding linear relationships, and even in predicting values within a data set. The example equation, \( y = -1.5x + 1 \), shows clearly that the line has a slope of -1.5 and a y-intercept of 1.
Graphing Linear Equations
Graphing a linear equation is a straightforward process once you understand the slope-intercept form. Start by identifying the y-intercept \( b \). This is your starting point on the y-axis. From there, use the slope \( m \) as a guide to determining the direction and angle of the line.Suppose we have the equation \( y = -1.5x + 1 \):
  • Begin by plotting the y-intercept: Place a point at \((0, 1)\).
  • Next, use the slope to find another point: The slope of -1.5 means for every step "right" along the x-axis, move 1.5 steps "down" on the y-axis. From \((0, 1)\), moving one unit to the right and 1.5 units down lands us at \((1, -0.5)\).
  • Repeat this process to plot additional points and draw a line through them, extended across the graph.
By connecting these points with a straight line, you accurately represent the linear equation. This approach provides both a visual and intuitive understanding of how changes in the slope or intercept affect the line's position on a graph.
Finding Slope and Y-Intercept
The ability to find the slope and y-intercept from a linear equation is essential for understanding and solving many algebra problems. By isolating these components in the equation, you can quickly determine how the line behaves.To find these components in a typical linear equation like \( y = mx + b \):
  • Slope \( (m) \): This is simply the coefficient in front of \( x \). In the equation \( y = -1.5x + 1 \), the slope is \( m = -1.5 \). This value indicates a line that falls from left to right.
  • Y-intercept \( (b) \): The y-intercept is the constant or the number without \( x \). For our example, \( b = 1 \). This means that the line crosses the y-axis at the point \((0, 1)\).
Once these values are identified, you can easily sketch the graph and gain insights into the line's real-world implications, such as predicting values or understanding relationships between variables.

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Most popular questions from this chapter

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