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Chapter 3: Problem 37

Graph each linear equation. \(y=2 x-5\)

Short Answer

Expert verified
The graph is a straight line passing through the points (0, -5) and (1, -3).

Step by step solution

01

Identify the Slope and Y-Intercept

The given equation is in the slope-intercept form, which is written as \(y = mx + b\). Here, \(m\) is the slope, and \(b\) is the y-intercept. For the equation \(y = 2x - 5\), the slope \(m\) is 2, and the y-intercept \(b\) is -5.
02

Plot the Y-Intercept

The y-intercept is the point where the line crosses the y-axis. In this case, the y-intercept is -5. Plot the point (0, -5) on the graph.
03

Use the Slope to Find Another Point

The slope of the line is 2, which means the rise over run is 2/1. From the y-intercept, move up 2 units and 1 unit to the right. Plot this second point at (1, -3).
04

Draw the Line

Using a ruler, draw a straight line through the two points (0, -5) and (1, -3). This line represents the graph of the equation \(y = 2x - 5\).

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Key Concepts

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

slope-intercept form
When you see a linear equation like the one provided in the exercise, it is important to recognize its form. The slope-intercept form of a linear equation is written as:

\(y = mx + b\)

Here:

  • \(m\) is the slope of the line
  • \(b\) is the y-intercept, which is where the line crosses the y-axis
Understanding the slope-intercept form helps you quickly identify the key characteristics of the line, like how steep it is and where it starts on the y-axis.
In our example equation, \(y = 2x - 5\), the slope \(m\) is 2 and the y-intercept \(b\) is -5. Knowing these values is the first step in graphing the line accurately.
y-intercept
The y-intercept is a crucial point in graphing linear equations. It's the point where your line will cross the y-axis. To find the y-intercept in the equation \(y = 2x - 5\), you look at the value of \(b\), which is -5 in this case.

This tells you that the line crosses the y-axis at (0, -5). Plot this point on your graph as it will be your starting point for drawing the line.

To ensure you've marked it correctly:

  • Find the y-axis on your graph
  • Move down to -5 on the y-axis
  • Mark that point (this is (0, -5))
Taking this step correctly makes the rest of the graphing much easier.
slope
The slope of a line describes its steepness and direction. In the slope-intercept form \(y = mx + b\), the slope is represented by the value \(m\). For our equation \(y = 2x - 5\), the slope \(m\) is 2.

To understand the slope, think of it as 'rise over run.鈥� This means:

  • 鈥楻ise鈥� is how much you go up or down
  • 鈥楻un鈥� is how much you go left or right
Here, a slope of 2 can be seen as '2/1', which means from any point on the line, you go up 2 units and then move 1 unit to the right.

From the y-intercept (0, -5), moving up 2 units and 1 unit to the right will get you to the next point at (1, -3). Plot this point on your graph.

Once you have these two points, draw a straight line through them, and you've successfully graphed the linear equation!

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