Problem 35 - Algebra 1

Question

Make a table of values for x = 1, 2, 3, and 4. Use the table to sketch a graph. Decide whether x and y vary directly or inversely. $$y=3 x$$

Step-by-Step Solution

Answer

The table of values is (x,y) = { (1,3), (2,6), (3,9), (4,12) }. The graph is a straight line passing through the origin, indicating a direct variation between x and y.

Step 1: Substitute x-values into the function

According to the function \( y = 3x \), we substitute the x-values 1, 2, 3, and 4 to get corresponding y-values.

Step 2: Create a table

From the above evaluation, the table of values will be: (x,y) = { (1,3), (2,6), (3,9), (4,12) }

Step 3: Sketch the graph

Plot these pairs of points on a graph. The plot shows a straight line passing through the origin.

Step 4: Determine the type of variation

In a directly proportional variation, as x increases, y also increases. This is exactly what is observed here, hence, x varies directly with y.

Key Concepts

Linear Equations

Linear equations are mathematical statements that show a direct relationship between two variables. Usually, you will find them in the format of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. A special case of linear equations is direct variation, expressed as \( y = mx \), where the line passes through the origin because the y-intercept \( b \) is zero. In these equations, any change in \( x \) leads to a proportional change in \( y \).

The linear equation given here, \( y = 3x \), perfectly illustrates this type of relationship. The coefficient \( 3 \) is the slope, indicating that for every unit increase in \( x \), \( y \) increases by 3 units. Knowing how to interpret and work with linear equations can help you predict how changes in one variable will affect another.

Graphing

Graphing linear equations is a powerful way to visually represent relationships between variables. By plotting points derived from a linear equation like \( y = 3x \), you can see the linear relationship between \( x \) and \( y \). The process involves placing points on a graph based on coordinates, like (1, 3), (2, 6), (3, 9), and (4, 12) in this case.

These points lie on a straight line. This confirms that the equation describes direct variation, and the line will pass through the origin. Here are a few things to remember about graphing linear equations:

• The slope determines how steep the line is. A larger slope value means a steeper line.
• The y-intercept is where the line crosses the y-axis. For direct variations, this point is always the origin, (0,0).
• Every point on the line satisfies the equation of the line.

Graphing is not just about plotting points; it's about understanding the story the graph tells about the relationship between the variables.

Tables of Values

Creating tables of values is a fundamental step in understanding and plotting linear equations. It involves substituting different values of \( x \) into the equation to find the corresponding \( y \) values. This is what we did for the linear equation \( y = 3x \).

For example, substituting \( x = 1 \) in the equation gives \( y = 3(1) = 3 \). Similarly, for \( x = 2 \), \( y = 6 \), and so on, creating the table of values (1, 3), (2, 6), (3, 9), (4, 12).

• These pairs represent coordinates that can be plotted on a graph.
• Visually, they help you understand the relationship between the two variables.
• A properly constructed table gives you a clear and organized view of how one variable changes in response to another.

By taking the time to create accurate tables of values, you establish a solid foundation for graphing linear equations and demonstrating direct variation.