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Chapter 3: Problem 24

Graph equation by hand. \(y=x\)

Short Answer

Expert verified
Plot points like (0, 0), (1, 1), (-1, -1), etc., and draw a straight line through them.

Step by step solution

01

Identify the Equation

The given equation is a linear equation: \(y = x\). This means for every value of \(x\), \(y\) takes the same value.
02

Create a Table of Values

Create a table listing several values of \(x\) and the corresponding values of \(y\). For example: \((x, y) = (0, 0), (1, 1), (2, 2), (-1, -1), (-2, -2)\).
03

Plot the Points

On a graph, plot the points from the table, such as \((0, 0)\), \((1, 1)\), \((2, 2)\), \((-1, -1)\), and \((-2, -2)\).
04

Draw the Line

Draw a straight line through all the plotted points. This line represents the graph of the equation \(y = x\).

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Key Concepts

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

linear equations
Linear equations are equations that create a straight line when plotted on a graph. They have the form y = mx + b, where m is the slope, and b is the y-intercept. In the case of the equation y = x, the slope m is 1, and the y-intercept b is 0.
This means that for each unit increase in x, y increases by the same amount. You can also imagine this as moving diagonally at a 45-degree angle since each x value aligns with an identical y value.
plotting points
Plotting points is a key skill in graphing linear equations. Each point represents a solution to the equation.
To plot a point, you need an (x, y) pair. For example, the point (1, 1) means that x = 1 and y = 1. On a graph, this pair is visualized by moving 1 unit to the right (for x) and 1 unit up (for y) from the origin (0,0).
Practicing plotting various points helps build an intuition for how an equation translates into a line.
table of values
Creating a table of values helps in visualizing the relationship between x and y.
To create a table for y = x, choose several values for x. Then calculate the corresponding y values. For instance:
  • if x = 0, then y = 0
  • if x = 1, then y = 1
  • if x = 2, then y = 2
  • and so on...
This table can be used to plot points on a graph accurately.
drawing graphs
Drawing graphs is the final step in visualizing linear equations. Once you have plotted all of your points, the next step is to draw a straight line through them. This line represents all the possible solutions to the equation.
Use a ruler to ensure the line is straight.
The line for y = x will go through points like (-2, -2), (-1, -1), (0, 0), (1, 1), and (2, 2). This confirms the relationship between x and y in the equation.

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