Problem 59 - Introductory Algebra for College Students

Question

Graph equation. \(3 y=9\)

Step-by-Step Solution

Answer

The graph of the equation \(3 y=9\) is a horizontal line parallel to the x-axis, passing through the point where \(y=3\).

Step 1: Solve the equation for y

First, divide both sides of the equation by 3 to isolate y. Doing so, you obtain \(y=3\). This means that for all x values, y will always equal to 3.

Step 2: Draw the Cartesian plane

Now, draw a Cartesian plane which includes the x and y axis. Numbers on the y axis should at least range from 0 to 3.

Step 3: Plot the graph

Finally, plot the line \$y=3\$ on the Cartesian plane. Since y is constant, the line drawn will be horizontal and parallel to the x-axis, passing through y=3 at all points.

Key Concepts

Solving Equations

Understanding how to solve equations is fundamental in graphing linear equations. When you're presented with something like \(3y = 9\), it's an equation that tells you there's a relationship between 'y' and the number 9, with 3 being a multiplier of 'y'. To find out what 'y' is, you need to perform operations that will isolate 'y' on one side. This usually involves dividing, adding, subtracting, or multiplying both sides of the equation by the same number to keep the equation balanced.

In this case, you divide both sides by 3 to get \(y = 3\). This simple process is at the heart of solving equations. Once you know 'y', you can graph this relationship on a Cartesian plane. To practice, try solving these equations: \(2x = 6\), \(4y + 4 = 12\), and \(x/2 - 1 = 2\).

Cartesian Plane

A Cartesian plane is a two-dimensional grid formed by the intersection of a vertical line (the y-axis) and a horizontal line (the x-axis). The point where they intersect is known as the origin, labeled as (0,0). The plane is divided into four quadrants, with positive and negative values on both axes. To plot a point, you locate the x value on the x-axis and match it with the y value on the y-axis.

If you're working with the equation \(y = 3\), every point will have a 'y' value of 3, regardless of what 'x' is. This tells you that the points you plot will align horizontally. A detailed understanding of the Cartesian plane is essential as it's not just a tool for plotting graphs but also for understanding functions, coordinates, and the overall 'map' of algebra.

Horizontal Line Graph

A horizontal line graph is a type of graph where all points have the same 'y' value and different 'x' values, creating a straight line that's parallel to the x-axis. For the equation \(y = 3\), it means that no matter what 'x' is, 'y' will always be 3. When you graph this equation, your horizontal line should pass through the 3 on the y-axis.

To visualize this, imagine walking along a straight path without going up or down any hills—that's like walking along a horizontal line on the Cartesian plane. Horizontal lines like these have a slope of zero because there is no vertical change as you move along the line.

Plotting Graphs

Plotting graphs is an exciting part of algebra that allows you to visualize equations and functions. Start by identifying the variables and their values, much like you did when you solved for 'y' in our example. You create a table of values where 'x' can be any number, but 'y' will always be 3 for our equation \(y = 3\). Then, on a Cartesian plane, plot points where the 'x' and 'y' values meet.

For example, point (0,3) is where 'x' is 0 and 'y' is 3. After plotting several points, connect them with a straight edge to form your graph. It's worth noting that the precision in plotting these points and drawing the line is crucial for an accurate representation of the equation. With these skills, you can graph any equation—and see the beauty of algebra come to life on paper!