Problem 49 - Algebra 1

Question

Use a table of values to graph the equation. \(x-2 y=6\)

Step-by-Step Solution

Answer

The graph of the equation \(x-2y=6\) is a straight line passing through points (-2,-4), (0,-3), (2,-2), and (4,-1).

Step 1: Rearrange the equation

First, let's rearrange the equation to solve for one variable. In this case, let's solve for y: \(x-2y=6\) rearranged, \(y = (x-6)/2\).

Step 2: Create a table of values

Create a table with two columns, one for x values and one for y values. Choose a sample of x-values. For simplicity, pick -2, 0, 2, 4. Substituting these x-values into \(y = (x-6)/2\) will get the corresponding y-values.

Step 3: Calculate y-values

Substitute each x-value into the equation to find the corresponding y-value. For x=-2: \(y = (-2-6)/2 = -4\). For x=0: \(y = (0-6)/2 = -3\). For x=2: \(y = (2-6)/2 = -2\). For x=4: \(y = (4-6)/2 = -1\).

Step 4: Plot the points and draw the graph

Plot the points on a graph. The points are: (-2,-4), (0,-3), (2,-2), and (4,-1). Draw a line through the points to complete the graph.

Key Concepts

Table of Values

The table of values is an essential tool for graphing linear equations. It's essentially a visual organizer, helping students to systemically plot the coordinates on a graph. To create one, draw a two-column table where one column represents the x-values and the other the y-values.

For the equation given in our exercise, \(x - 2y = 6\), we want to rearrange it to solve for y which is our dependent variable. Once rearranged, y becomes the output for each corresponding x input we choose. By selecting a variety of x-values, such as -2, 0, 2, and 4, we can solve for the y-values and populate our table. It's important to choose a range of x-values to get the full shape of the graph.

Linear Equation Graphing

The process of graphing a linear equation involves plotting points represented by pairs of x and y coordinates and connecting them to form a straight line. The slope and y-intercept derived from the equation give shape to the graph.

After creating your table of values and determining several coordinate pairs, you plot them onto a Cartesian plane. Each pair represents a point where the line defined by the equation crosses that exact combination of x and y. When we graph the equation from this exercise, \(y = \frac{x-6}{2}\), each point plots out the visualization of the relationship between the two variables, showing how y changes as x varies.

Solving for y

The methodology of solving for y in a linear equation like \(x - 2y = 6\) is a crucial step in graphing. This turns the standard form of a linear equation into slope-intercept form, \(y = mx + b\), which tells us the slope of the line (m) and where it crosses the y-axis (b).

To isolate y, we perform algebraic operations to get \(y\) on one side by itself. As done in the exercise, we subtract \(x\) and then divide by -2, resulting in the equation \(y = \frac{x-6}{2}\). With this equation, you can plug in any value for x to calculate the corresponding y—the foundation for building our table of values.

Plotting Points on a Graph

Lastly, the skill of plotting points on a graph is what brings the visual representation of a linear equation to life. Once you have your table of values with corresponding x and y pairs, you plot them on the Cartesian plane. Each point is plotted at the intersection that corresponds to its x (horizontal axis) and y (vertical axis) values.

For our example, points like \( (-2, -4), (0, -3), (2, -2), (4, -1) \) are marked on the graph. After plotting the points, you can draw a straight line through them to complete the graph, which is the visual representation of the equation \(x - 2y = 6\). Remember to ensure that the points are accurate and that the line extends across the graph to illustrate the infinite number of solutions that satisfy the linear equation.