Problem 71 - 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. $$y=\frac{1}{3} x+1$$

Step-by-Step Solution

Answer

The solutions for the equation \(y=\frac{1}{3}x+1\) are (-2,-1/3), (-1,2/3), (0,1), (1,4/3), and (2,5/3).The graph is a straight line starting from (0,1) with a slope of 1/3.

Step 1: Understanding the Equation

The given linear equation is \(y=\frac{1}{3} x+1\). The form of this equation is \(y=mx+c\), where \(m\) and \(c\) stand for the slope and the y-intercept, respectively. In this case, the slope \(m\) is \(\frac{1}{3}\) and the y-intercept \(c\) is \(1\). This means that the line starts at \(1\) on the y-axis and increases by \(\frac{1}{3}\) for each increment of \(x\).

Step 2: Creating a Table of Values

A table of values can be created to find possible solutions for the equation. By substituting different values of \(x\) into the equation, corresponding \(y\) values can be calculated. Here are five possible solutions (pairs of x and y values) after performing the substitutions: For \(x=-2, y = -\frac{1}{3}\); For \(x=-1, y = \frac{2}{3}\); For \(x=0, y = 1\); For \(x=1,y = \frac{4}{3}\); For \(x=2, y = \frac{5}{3}\).

Step 3: Graphing the Equation

The graph of the equation can be plotted using the solutions found in the table. Start by placing a point at the y-intercept (0,1). Then, plot the points corresponding to the solutions in the table. Connect the points to form a straight line. The graph of any linear equation will always be a straight line.

Key Concepts

Graphing Linear Equations

Graphing linear equations is an essential skill in algebra which involves plotting a straight line on a coordinate plane. The line represents all the solutions to the given linear equation. To graph a linear equation like \(y = \frac{1}{3}x + 1\), begin by identifying important components such as the slope and the y-intercept. These elements help you understand how the line will appear on the plot.
Graphing provides a visual representation of an equation and helps to understand the relationship between variables. Once you have the necessary points, you draw the line by connecting them in sequence, illustrating that many solutions satisfy the equation, not just the initially plotted points.

Slope-Intercept Form

The slope-intercept form is a linear equation format written as \(y = mx + c\). Here, \(m\) represents the slope of the line, and \(c\) indicates where the line crosses the y-axis, known as the y-intercept. For the equation \(y = \frac{1}{3}x + 1\), the slope \(\frac{1}{3}\) tells us how steep the line is.
Each time \(x\) increases by 1, \(y\) increases by one-third. The y-intercept is 1, so the line crosses the y-axis where \(y = 1\).

• **Slope**: Shows the direction and steepness of a line
• **Y-intercept**: The starting point where the line crosses the y-axis.

Understanding the slope-intercept form is crucial as it offers quick information about the line's direction and starting point.

Table of Values

A table of values is an organized list of specific solutions for a linear equation. It involves choosing various \(x\) values and calculating the corresponding \(y\) values according to the equation. This method is helpful for determining points that will later be plotted on a graph.
For the equation \(y = \frac{1}{3}x + 1\), you can select convenient values for \(x\) such as -2, -1, 0, 1, and 2. Calculating for each:

• When \(x = -2\), \(y = -\frac{1}{3}\)
• When \(x = -1\), \(y = \frac{2}{3}\)
• When \(x = 0\), \(y = 1\)
• When \(x = 1\), \(y = \frac{4}{3}\)
• When \(x = 2\), \(y = \frac{5}{3}\)

These pairs become the points that can be plotted to graph the equation. Creating a robust table of values is a foundational step to ensure accurate graphing.

Plotting Points

Plotting points on a graph involves marking coordinates on a grid following the pairs calculated from the table of values. Each pair includes \(x\) and \(y\) values which act as coordinates on the coordinate plane.
To plot the linear equation \(y = \frac{1}{3}x + 1\), you start by plotting the y-intercept at \((0, 1)\). Then, you mark points like \((-2, -\frac{1}{3})\), \((-1, \frac{2}{3})\), \((1, \frac{4}{3})\), and \((2, \frac{5}{3})\).

• Ensure that each point is accurately placed according to its coordinates.
• Once all points are plotted, connect them with a straight line using a ruler.

This straight line represents all solutions to the equation, demonstrating how the \(y\) values change with \(x\). Proper plotting is vital for creating a precise graph that visually expresses the equation's solutions.