Problem 5 - Elementary Algebra

Question

The relationship of \(x\) and \(y\) is a direct variation. When \(x=2, y=6\). 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 \(x=5\).

Step-by-Step Solution

Answer

a. \(k = 3\) b. \(y = 3x\) c. \(y = 12\) d. Graph the line \(y = 3x\). e. From the graph, \(y = 15\) at \(x = 5\).

Step 1: Find the constant of proportionality

In direct variation, the relationship between two variables can be expressed as \(y = kx\). Given \(x = 2\) and \(y = 6\), plug these values into the equation to find the constant of proportionality, \(k\). \[ 6 = k \cdot 2 \] Solve for \(k\) by dividing both sides by 2: \[ k = \frac{6}{2} = 3 \]

Step 2: Write the equation of direct variation

Now that the constant of proportionality, \(k\), is known, substitute \(k = 3\) back into the equation \(y = kx\). Therefore, the equation representing the direct variation is: \[ y = 3x \]

Step 3: Find \(y\) when \(x = 4\)

Use the direct variation equation \(y = 3x\) to find \(y\) when \(x = 4\). Substitute \(x = 4\) into the equation: \[ y = 3 \cdot 4 = 12 \]

Step 4: Graph the equation using slope-intercept form

The equation \(y = 3x\) can be graphed using the slope-intercept form \(y = mx + b\). Here, the slope \(m = 3\) and the y-intercept \(b = 0\). Plot the y-intercept \((0, 0)\) and use the slope to find another point. From \((0, 0)\), move up 3 units and to the right 1 unit to plot the point \((1, 3)\). Draw a straight line through these points.

Step 5: Use the graph to find \(y\) when \(x = 5\)

Using the graph of the equation \(y = 3x\), locate \(x = 5\) on the x-axis. Follow the corresponding vertical line to where it intersects the graph. The point of intersection is \( (5, 15) \). Thus, when \(x = 5\), \(y = 15\).

Key Concepts

constant of proportionality

When two variables, like \(x\) and \(y\), vary directly, their relationship can be expressed as \(y = kx\). Here, \(k\) is known as the constant of proportionality. It is the factor that relates how much one variable changes in response to the other. In the given exercise, we are told that when \(x = 2\), \(y = 6\). By substituting these values into the equation, we get:

\[ 6 = k \times 2 \]

To solve for \(k\), divide both sides by 2:

\[ k = \frac{6}{2} = 3 \]
This tells us that every time \(x\) changes, \(y\) will change three times faster.

direct variation equation

Once the constant of proportionality \(k\) is found, it can be used to write the direct variation equation. If we know that \(k = 3\), we substitute this back into the formula \(y = kx\), resulting in:

\[ y = 3x \]
This is our direct variation equation. It shows the straight-line relationship between \(x\) and \(y\) where \(y\) changes by a factor of 3 for every unit increase in \(x\). You can use this equation to find \(y\) easily, as shown in the next step when we found \(y = 12\) for \(x = 4\).

slope-intercept form

The equation of a line in slope-intercept form is \(y = mx + b\). Here, \(m\) represents the slope, and \(b\) represents the y-intercept. For the equation \(y = 3x\), the slope \(m\) is 3 and the y-intercept \(b\) is 0. This means that the line crosses the origin (0, 0) and rises 3 units for every 1 unit it goes to the right.

To graph the line, start at the y-intercept (0, 0). From there, use the slope to find another point: go up 3 units and over 1 unit to the right to plot (1, 3). Drawing a straight line through these points will give you the graph of the equation.

graphing linear equations

Graphing linear equations helps visualize the relationship between the variables. For the equation \(y = 3x\), you start with the y-intercept at the origin (0, 0). Using the slope of 3, which tells you to go up 3 units for every 1 unit to the right, you can plot your next point at (1, 3). Continuing this pattern helps you draw the line accurately.

When graphing, if you need to find \(y\) for a specific value of \(x\), you can simply look at the graph. For instance, to find \(y\) when \(x = 5\), you locate 5 on the x-axis and trace vertically until you hit the graphed line. At this intersection, the coordinate point (5, 15) tells you that \(y\) equals 15.