Problem 72 - 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 graph of the equation \(y = \frac{1}{3}x - 1\) can be drawn by plotting the points \((-2, -\frac{5}{3}), (-1, -\frac{4}{3}), (0, -1), (1, -\frac{2}{3}), (2, -\frac{1}{3})\) and drawing a line through them.

Step 1: Make a table of values

Create a table of values. Let \(x = -2, -1, 0, 1, 2\). Now, substitute each value of \(x\) into the linear equation and solve for \(y\).

Step 2: Calculate y values

Substitute the \(x\) values into the equation: \nFor \(x=-2, y = \frac{1}{3}(-2) - 1 = -\frac{5}{3} \nFor \(x=-1, y = \frac{1}{3}(-1) - 1 = -\frac{4}{3} \nFor \(x=0, y = \frac{1}{3}(0) - 1 = -1 \nFor \(x=1, y = \frac{1}{3}(1) - 1 = -\frac{2}{3} \nFor \(x=2, y = \frac{1}{3}(2) - 1 = -\frac{1}{3}

Step 3: Plot the points on the graph

Plot the points from your table on the graph. The points will be \((-2, -\frac{5}{3}), (-1, -\frac{4}{3}), (0, -1), (1, -\frac{2}{3}), (2, -\frac{1}{3})\)

Step 4: Draw the line

Draw a straight line through the plotted points to graph the linear equation.

Key Concepts

Graphing Coordinates

Graphing coordinates involves placing points on a grid to visualize relationships in equations. This is an essential technique in algebra for understanding patterns and behaviors of linear equations. You begin with an equation, like the one given: \( y = \frac{1}{3}x - 1 \).
Each ordered pair \((x, y)\) represents a coordinate on the graph:

• The first number in the pair is the x-coordinate, which tells you how far to move horizontally from the origin.
• The second number is the y-coordinate, which tells you how far to move vertically.

To plot a point, start from the origin (0,0). Move to the x-value on the horizontal axis, then shift up or down to align with the y-value on the vertical axis. When plotting the points from the solution, such as \((-2, -\frac{5}{3})\), you start at \(x = -2\) and move to \(y = -\frac{5}{3}\), marking that point. Connecting these points will show the line defined by the equation. This visual representation helps to understand the equation's slope and intercept.

Table of Values

Creating a table of values is a straightforward way to organize the x and y values that satisfy your equation. It's like a roadmap for your graph! With the equation \( y = \frac{1}{3}x - 1 \), choose values for \(x\) that are manageable.
In this exercise, the chosen x-values are \(-2, -1, 0, 1, 2\). For each of these x-values, you substitute into the equation to find the y-value:

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

Once you have completed the table, plot these points on the graph. This method efficiently provides a visual confirmation of how points align to form a line, revealing the linear relationship between x and y.

Slope-Intercept Form

The slope-intercept form is a handy way to quickly identify the properties of a linear equation. It is written as \( y = mx + b \). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept.
In our example with the equation \( y = \frac{1}{3}x - 1 \):

• The slope \(m\) is \(\frac{1}{3}\). This indicates that for every 3 units moved horizontally, the line rises 1 unit vertically. A positive slope means the line angles upwards to the right.
• The y-intercept \(b\) is \(-1\), showing where the line crosses the y-axis. It's the value of \(y\) when \(x = 0\), which confirms the point \((0, -1)\).

Understanding this form makes it easy to graph equations, as you can start by plotting the y-intercept and then use the slope to find subsequent points. This forms the backbone of discussing linear equations and heavily contributes to comprehending their graphical representations.