Problem 65 - 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. $$y=-3 x-1$$

Step-by-Step Solution

Answer

Five solutions of the equation \(y=-3x-1\) include: (-2,5), (-1,2), (0,-1), (1,-4), and (2,-7), obtained by substituting each \(x\) value into the equation and solving for \(y\). Your graph should show these points connected by a straight line.

Step 1: Understand the Equation

The equation \(y = -3x - 1\) is a linear equation in two variables, \(x\) and \(y\). It represents a straight line on a graph.

Step 2: Create a Table of Values

Start by choosing five different values for \(x\). Any values may elect, but often simple numbers like -2, -1, 0, 1, 2 are chosen because they're easy to work with. Then, substitute each \(x\) value into the equation to find the corresponding \(y\) value to get five ordered pairs (x, y). This forms the table of values.

Step 3: Find the Solutions

Substitute the chosen \(x\) values one by one into the equation. For example, if \(x = -2\), \(y = -3*-2 - 1 = 5\). Continue this process for all \(x\) values chosen to get the five ordered pairs.

Step 4: Graph the Linear Equation

Plot the ordered pairs (x, y) that obtained on the graph. Connect the plotted points with a straight line to get the graph for the equation \(y = -3x - 1\).

Step 5: Interpret the Graph

Each point on the straight line represents a solution for the equation \(y = -3x - 1\), and the direction of the line indicates that as \(x\) increases, \(y\) decreases since the slope is negative.

Key Concepts

Graphing Linear Equations

Graphing linear equations means drawing the picture of an equation that forms a straight line on a Cartesian plane. The equation given here is \( y = -3x - 1 \). When you graph a linear equation, you show all the solutions of the equation visually. Each point on the line is a solution because it makes the equation true. Using the graph, you can easily identify how changes in the value of \( x \) affect the value of \( y \). In this equation, the negative sign in front of 3 indicates the line will slope downwards from left to right.

Table of Values

Creating a table of values is a straightforward approach to get several solutions for a linear equation quickly. Here, you select different values for \( x \), substitute them into the equation \( y = -3x - 1 \), and solve for \( y \). For example:

• If \( x = -2 \), then \( y = 5 \)
• If \( x = -1 \), then \( y = 2 \)
• If \( x = 0 \), then \( y = -1 \)
• If \( x = 1 \), then \( y = -4 \)
• If \( x = 2 \), then \( y = -7 \)

The table of values will help us later to plot the points on the graph. This approach is useful because it provides multiple solutions, allowing for accurate graph plotting.

Solutions of Linear Equations

Each solution of a linear equation in two variables corresponds to an ordered pair \( (x, y) \). A solution satisfies that specific equation, meaning if you plug \( x \) and \( y \) into the equation, it holds true.Taking \( y = -3x - 1 \), let's verify one of our solutions: \( (x, y) = (0, -1) \).Substitute \( x = 0 \):\[ y = -3(0) - 1 = -1 \]The equation is true, so \( (0, -1) \) is indeed a solution. Similarly, all the points used in the table of values can be confirmed this way. Solutions represent all the ordered pairs on the graph.

Ordered Pairs

Ordered pairs are coordinates written in the form \( (x, y) \). They indicate the position of a point on a graph. In linear equations, each ordered pair is a potential solution to the equation.Consider these ordered pairs from the equation \( y = -3x - 1 \):

• \( (-2, 5) \)
• \( (-1, 2) \)
• \( (0, -1) \)
• \( (1, -4) \)
• \( (2, -7) \)

Each pair can be plotted on a graph, creating a visual representation of the linear equation. Ordered pairs are the foundation of graphing; without them, it would be impossible to draw accurate graphs. They show the relationship between \( x \) and \( y \) clearly.