Problem 7 - Elementary Algebra

Question

The relationship of \(x\) and \(y\) is a direct variation. When \(x=1, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this direct variation. c. Find \(y\) when \(x=4\), d. Use slope-intercept graphing to graph this equation. e. Use the graph to find \(y\) when \(x=2\).

Step-by-Step Solution

Answer

\(k = 6\), equation is \(y = 6x\), at \(x = 4\), \(y = 24\), and at \(x = 2\), \(y = 12\).

Step 1: Identify the formula for direct variation

A direct variation relationship between two variables can be written as \(y = kx\), where \(k\) is the constant of proportionality.

Step 2: Identify given values

Given \(x = 1\) and \(y = 6\). Using the direct variation formula, substitute these values into the equation \(y = kx\).

Step 3: Solve for the constant of proportionality

Substitute \(x = 1\) and \(y = 6\) into the equation: \(6 = k \times 1\) Solve for \(k\): \(k = 6\).

Step 4: Write the direct variation equation

The direct variation equation representing the relationship between \(x\) and \(y\) is: \(y = 6x\).

Step 5: Calculate \(y\) for a given \(x = 4\)

Substitute \(x = 4\) into the equation \(y = 6x\): \(y = 6 \times 4 = 24\).

Step 6: Set up graphing parameters for the equation

To use slope-intercept graphing, note that the equation is in the form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept. Here, \(y = 6x\) means the slope \(m = 6\) and \(b = 0\).

Step 7: Graph the equation

Start by plotting the y-intercept (0,0) on the graph. Next, use the slope \(6\) to plot the next point (1,6) by rising 6 units up and running 1 unit right. Draw the line through these points.

Step 8: Use the graph to find \(y\) when \(x = 2\)

Locate \(x = 2\) on the graph. From \(x = 2\), go up vertically until intersecting the line. Check the corresponding \(y\)-value, which corresponds to \(y = 12\).

Key Concepts

constant of proportionality

The concept of 'constant of proportionality' is foundational to understanding direct variation. When two quantities vary directly, one is a constant multiple of the other. This constant multiple is known as the 'constant of proportionality' and is frequently denoted as \(k\).
For example, in the given exercise, we have a direct variation relationship between \(x\) and \(y\). Given \(x = 1\) and \(y = 6\), we use the formula \(y = kx\). Substituting the given values:

• \[ 6 = k \times 1 \] results in \[ k = 6 \]

This means that \(k\) or the constant of proportionality is 6 in this case. Understanding this concept helps in quickly determining relationships in many mathematical problems.

slope-intercept form

The slope-intercept form of a linear equation is a powerful tool for graphing and analyzing linear relationships. It is generally written as \[ y = mx + b \], where

• \(m\): represents the slope of the line
• \(b\): is the y-intercept, where the line crosses the y-axis

In our exercise, the direct variation equation is \[ y = 6x \]. This is already in slope-intercept form with \(m = 6\) and \(b = 0\).
The slope \(m = 6\) tells us that for every unit increase in \(x\), \(y\) increases by 6 units. The y-intercept \(b = 0\) indicates that the line passes through the origin (0,0).
By identifying these components, we can easily graph and analyze the behavior of linear equations.

graphing linear equations

Graphing linear equations is a valuable skill in understanding and interpreting relationships in algebra. We use the slope-intercept form to make this process simpler.
From our exercise, the equation \[ y = 6x \] has a slope \(m = 6\) and y-intercept \(b = 0\). Here’s how to graph it:

• Start by plotting the y-intercept, which is (0,0).
• Use the slope to determine the next points. The slope of 6 means we rise 6 units up and run 1 unit right from the y-intercept to get the point (1,6).
• Draw a line through these points to represent the equation.

This graph visually demonstrates the relationship between \(x\) and \(y\). By looking at the graph, we can find unknown values easily. For instance, to find \(y\) when \(x = 2\), locate \(x = 2\) on the x-axis, then move vertically to the line and find the corresponding \(y\)-value, which in this case is \[ y = 12 \].
Graphing not only helps in solving problems but also in better understanding how variables interact with one another.