Problem 72 - 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. $$y=\frac{1}{3} x-1$$

Step-by-Step Solution

Answer

The graph of the equation \(y = \frac{1}{3}x - 1\) forms a straight line with slope \(\frac{1}{3}\) and y-intercept -1. The five points plotted are (-6,-3), (-3,-2), (0,-1), (3,0), (6,1).

Step 1: Identifying the Slope and Y-intercept

The slope-intercept form of a linear equation is \(y = mx + b\), where m is the slope and b is the y-intercept. The given equation is \(y = \frac{1}{3}x - 1\), therefore, the slope \(m = \frac{1}{3}\) and the y-intercept \(b = -1\).

Step 2: Constructing a Table of Values

Choose five x-values and substitute them into the equation to find the corresponding y-values. For example, for \(x = -6, -3, 0, 3, 6\), respectively, the y-values will be \(-3, -2, -1, 0, 1\).

Step 3: Plotting the Points

On a graph, plot the points calculated in the table of values. So, coordinate points to be plotted are (-6,-3), (-3,-2), (0,-1), (3,0), (6,1).

Step 4: Drawing the Line

Connect the points on the graph paper. Since this is a linear equation, all points should form a straight line.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is vital to solving and graphing equations effectively. The formula is given by \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) indicates the y-intercept.

• The **slope** \(m\) determines the tilt or steepness of the line. It is the ratio of the vertical change (rise) over the horizontal change (run) between any two points on the line.
• The **y-intercept** \(b\) is the point where the line crosses the y-axis. This happens when the value of \(x\) is zero.

For instance, in the equation \(y = \frac{1}{3}x - 1\), the slope is \(\frac{1}{3}\), which means that for every 3 units you move horizontally, the line moves 1 unit vertically. The y-intercept is \(-1\), crossing the y-axis at the point \((0, -1)\). This format helps in quickly sketching the graph of the equation.

Graphing Equations

Graphing equations involves creating a visual representation of solutions to an equation on a coordinate plane. A linear equation like \(y = \frac{1}{3}x - 1\) appears as a straight line on the graph.
The process includes several steps:

• Identify the slope and y-intercept using the slope-intercept form \(y = mx + b\).
• Use this information to plot the starting point of the line (the y-intercept) on the graph.
• Use the slope to find additional points, moving right for positive slopes and left for negative slopes, while adjusting the vertical position accordingly.

Ensure you mark these points carefully before drawing the line. Connecting these points with a straight line completes the graph of the linear equation. This line represents all potential solutions to the equation.

Table of Values

When graphing, constructing a table of values simplifies the process by organizing x-values with their corresponding y-values. This table acts as a roadmap to plot accurate points on the graph.
Here's how to go about it:

• Select some values for \(x\) within your graph's scale (try a mix of negative, zero, and positive numbers for diversity).
• Substitute each chosen \(x\)-value into the linear equation to calculate the respective \(y\)-value.
• Record these \((x, y)\) pairs in a list or table.

For example, choosing \(x = -6, -3, 0, 3,\) and \(6\) for our equation \(y = \frac{1}{3}x - 1\):

• For \(x = -6\), \(y = -3\)
• For \(x = -3\), \(y = -2\)
• For \(x = 0\), \(y = -1\)
• For \(x = 3\), \(y = 0\)
• For \(x = 6\), \(y = 1\)

Plotting these points next helps in visualizing the graph accurately.

Plotting Points

Plotting points is the action of placing coordinates on a graph based on values in a table. It transforms the abstract equation into a visual line.
Follow these simple steps:

• Begin with the y-intercept, the starting point. For the equation \(y = \frac{1}{3}x - 1\), start at \((0, -1)\).
• Proceed by using the slope to find new points. For instance, from \((0, -1)\), move right by 3 units and up by 1 unit to \((3, 0)\).
• Refer back to the table of values and mark each \((x, y)\) point on the graph paper or screen.

By plotting points such as \((-6, -3)\), \((-3, -2)\), \((0, -1)\), \((3, 0)\), and \((6, 1)\), you'll visibly see the line forming.
This line represents all possible solutions to the equation, demonstrating how y changes as x changes.