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Chapter 3: Problem 22

Graph each linear equation. Plot four points for each line. $$y=x-1$$

Short Answer

Expert verified
Plot points (0, -1), (1, 0), (2, 1), and (-1, -2). Draw a line through them.

Step by step solution

01

Understand the Equation

The given equation is a linear equation in the form of \(y = mx + b\). Here, \(m = 1\) (the slope) and \(b = -1\) (the y-intercept). This means the line has a slope of 1 and intersects the y-axis at -1.
02

Plot the Y-Intercept

Start by plotting the y-intercept. Since the y-intercept is -1, plot the point (0, -1) on the graph.
03

Determine Additional Points

Using the slope 1 (which means 'rise over run' = 1/1), find additional points. For each step right (increase x by 1), move up one step (increase y by 1).
04

Calculate and Plot Points

Choose three more values of x and calculate their corresponding y values.- For \(x = 1\): \(y = 1 - 1 = 0\) (plot point (1,0))- For \(x = 2\): \(y = 2 - 1 = 1\) (plot point (2,1))- For \(x = -1\): \(y = -1 - 1 = -2\) (plot point (-1,-2))
05

Draw the Line

Once the points (0, -1), (1, 0), (2, 1), and (-1, -2) are plotted, draw a straight line through them. This is the graph of the equation \(y = x - 1\).

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Key Concepts

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

linear equation
A linear equation is one that forms a straight line when graphed on a coordinate plane. The most common form of a linear equation is \[ y = mx + b \] where:
  • **y** is the dependent variable, usually plotted on the vertical axis.
  • **x** is the independent variable, usually plotted on the horizontal axis.
  • **m** is the slope, which measures the steepness and direction of the line.
  • **b** is the y-intercept, where the line crosses the y-axis.

In the given exercise, we have the linear equation \[ y = x - 1 \]. This fits the standard form, with **m = 1** and **b = -1**.
slope
The slope of a line is a measure of how much the y-value changes for every change in the x-value. It's often referred to as 'rise over run'.

In our equation \[ y = x - 1 \], the slope **m** is equal to 1. This means for every 1 unit increase in *x*, *y* also increases by 1 unit.

To visualize this:
  • Move 1 step to the right (positive x direction).
  • Move 1 step up (positive y direction).

This consistent rate of change is crucial in determining the line's direction and steepness in the graph.
y-intercept
The y-intercept is the point where the line crosses the y-axis. For the equation \[ y = x - 1 \], the y-intercept **b** is -1.

This means the line crosses the y-axis at the point (0, -1). To plot it on the graph:
  • Start at the origin (0,0).
  • Move down one step to (0, -1).

By plotting this point, you set a foundation to draw the rest of the line using the slope.
plot points
Plotting points is a method to represent the values of the linear equation on a graph. For \[ y = x - 1 \], we need to find corresponding y-values for different x-values.

For example:
  • When *x* = 0, \[ y = 0 - 1 = -1 \] so plot (0, -1).
  • When *x* = 1, \[ y = 1 - 1 = 0 \] so plot (1, 0).
  • When *x* = 2, \[ y = 2 - 1 = 1 \] so plot (2, 1).
  • When *x* = -1, \[ y = -1 - 1 = -2 \] so plot (-1, -2).

After plotting, draw a straight line through these points to represent the equation visually.

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Most popular questions from this chapter

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