Problem 61 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=2 x+1$$

Step-by-Step Solution

Answer

The five solutions for the equation \(y = 2x + 1\) are: \[(-2, -3), (-1, -1), (0, 1), (1, 3), (2, 5)\], and the line that passes through these points gives the graph of the equation.

Step 1: Understand the equation

We have a linear equation of the form \(y = mx + b\), where \(m = 2\) is the slope and \(b = 1\) is the y-intercept. For every increase of \(1\) unit in \(x\), \(y\) will increase by \(2\) units.

Step 2: Create a table of values

Start by choosing arbitrary values for \(x\). Let's choose \(-2, -1, 0, 1, 2\) for \(x\) as our five solutions.

Step 3: Calculate corresponding y-values

Substitute each x-value into the equation to find the corresponding y-value. This will give us our solutions: \[(-2, -3), (-1, -1), (0, 1), (1, 3), (2, 5)\]

Step 4: Plot the points

Plot each of these points on a graph. As the slope is positive, the line slants upwards.

Step 5: Draw the line

Then draw a straight line through all the points. This line represents all possible solutions for the equation

Key Concepts

Graphing Lines

Graphing lines helps us visually represent solutions of a linear equation. It involves plotting points on a graph that a line passes through. When a linear equation is graphed, it usually takes the form of a straight line.
Understanding how to graph these lines makes it easier to predict and visualize relationships between two variables.
Each point on the line is a solution to the linear equation, indicating a pair of values that satisfy the equation together. Creating graphs is an essential skill in algebra and mathematics in general, as it allows us to analyze and make decisions based on data.

Slope-Intercept Form

The slope-intercept form is a popular way to express linear equations. It is given by the formula: \[ y = mx + b \]Where:

• \(m\) is the slope of the line. The slope represents the rate of change, or how much \(y\) changes when \(x\) increases by 1.
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

This format makes it easy to quickly identify the slope and the y-intercept.
Understanding the slope-intercept form is crucial as it simplifies the process of graphing a line from its equation. By knowing \(m\) and \(b\), you can instantly predict the line's direction and vertical starting point on a coordinate plane.

Table of Values

A table of values is a useful tool to find different solutions or coordinate pairs for a linear equation. It involves choosing several values for \(x\) and calculating the corresponding \(y\) values using the equation.
Here’s how to create a table of values for the equation \(y = 2x + 1\):

• Select different values for \(x\) such as -2, -1, 0, 1, and 2.
• Substitute these \(x\) values into the equation to find \(y\).
• List the resulting \(x, y\) pairs, creating a table like this:
  • -2 ➔ -3
  • -1 ➔ -1
  • 0 ➔ 1
  • 1 ➔ 3
  • 2 ➔ 5

Utilizing a table of values simplifies the process of plotting points on a graph, which in turn helps in visualizing the linear relationship represented by the equation.

Coordinate Plane

The coordinate plane is a two-dimensional surface formed by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
Each point on this plane is represented by an ordered pair \((x, y)\), indicating its horizontal and vertical position, respectively.
Using the coordinate plane is essential for plotting points, graphing lines, and visualizing mathematical relationships. For example, when you plot the points \((-2, -3), (-1, -1), (0, 1), (1, 3), (2, 5)\) on a coordinate plane, you will connect them to form a straight line. This line graphically represents the solutions of the linear equation. Graphing on a coordinate plane helps provide a clear and visual understanding of mathematical concepts and relationships.