Problem 64 - Introductory Algebra for College Students

Question

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=-x+3$$

Step-by-Step Solution

Answer

The five solutions from the table of values are (0, 3), (-1, 4), (1, 2), (2, 1), (-2, 5). The graph is a straight line that intersects these points.

Step 1: Write down the equation

The equation that needs to be graphed is \(y=-x+3\)

Step 2: Create a table of values

The table of values is used to show the x and y coordinates that satisfy the equation. Choose a few values for the variable \(x\), then substitute each value into the equation to find the corresponding \(y\) value.

Step 3: Fill the table with values following the relation \(y=-x+3\)

For example, if \(x=0\), then \(y=-0+3=3\). Now follow this process for various x values. With \(x=-1, x=1, x=2, x=-2\), we get \(y=4, y=2, y=1, y=5\) respectively.

Step 4: Graph the points obtained from the table of values

Take each pair (x, y) from the table and plot it on the graph in their respective positions. Connect the dots with a straight line, which is the graphical representation of the equation.

Key Concepts

Understanding Table of Values

A table of values is an essential tool that helps us to plot graphs of linear functions. It is simply a collection of ordered pairs

• Each pair corresponds to one solution of the equation
• Ordered pairs are in the form \((x, y)\)

by selecting different numbers for the 'x' value, we can calculate the 'y' value.
Substitute each 'x' value into the linear equation, like in our exercise: \(y = -x + 3\).
For example, if you choose an 'x' value of \(2\), then substitute it into the equation to get \(y = -2 + 3\), which equals \(1\). Thus, \((2, 1)\) is one point in the table.
Once you have calculated multiple points, you can fill out a comprehensive table of values. This gives you handy points to plot on a coordinate system, aiding in drawing the linear graph.

Navigating the Coordinate System

The coordinate system is a crucial framework used in graphing. It consists of two axes:

• The horizontal axis known as the x-axis
• The vertical axis known as the y-axis

These axes intersect at a point called the origin (0,0).
The coordinate system allows for the location of points based on ordered pairs \((x, y)\). Each value in the pair determines the position:

• The 'x' value shows the position along the horizontal axis
• The 'y' value shows the position along the vertical axis

In practical terms, to plot the point \((-1, 4)\) on our coordinate system:

• Move left 1 unit from the origin along the x-axis
• Then up 4 units along the y-axis

By plotting all the points from the table of values, you form a straight line on the graph, representing the linear equation.

Exploring Linear Functions

Linear functions are a fundamental part of mathematics, describing a constant rate of change. These functions are typically expressed in the form \(y = mx + c\), where:

• 'm' is the slope, showing the rate at which 'y' changes with respect to 'x'
• 'c' is the y-intercept, the point where the line crosses the y-axis

The equation \(y = -x + 3\) is an example of a linear function.
The slope here is \(-1\), indicating the line descends when moving from left to right.
The y-intercept is \(3\), meaning the line crosses the y-axis at point \(3\). Linear functions graph as straight lines, making them easier to visualize and work with.