Problem 10 - Elementary Algebra

Question

The relationship of the amount of salad dressing, \(x\), and the amount of sodium in the dressing, \(y\), is a direct variation. Six servings of dressing contain \(1800 \mathrm{mg}\) of sodium. a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the amount of sodium in a bottle that contains 16 servings of salad dressing. d. Use slope-intercept graphing to graph this equation. e. Use the graph to find the amount of sodium in 3 servings of salad dressing.

Step-by-Step Solution

Answer

The constant of proportionality is 300 mg per serving, the equation is \( y = 300x \), the sodium in 16 servings is 4800 mg, and there are 900 mg of sodium in 3 servings.

Step 1: Understand the relationship

Recognize that the relationship between the amount of salad dressing, \( x \), and the amount of sodium, \( y \), is a direct variation. This means that \( y = kx \), where \( k \) is the constant of proportionality.

Step 2: Find the constant of proportionality \( k \)

We know that 6 servings contain 1800 mg of sodium. From the direct variation formula \( y = kx \), we can set \( y = 1800 \) mg and \( x = 6 \) servings. Solving for \( k \) gives:\[ k = \frac{y}{x} = \frac{1800 \text{ mg}}{6 \text{ servings}} = 300 \frac{\text{mg}}{\text{serving}} \]

Step 3: Write the equation representing the relationship

Using the found value of \( k \), the equation representing the direct variation relationship is:\[ y = 300x \]

Step 4: Calculate the amount of sodium in 16 servings

Substitute \( x = 16 \) servings into the equation \( y = 300x \) to find the amount of sodium:\[ y = 300 \times 16 = 4800 \text{ mg} \]

Step 5: Graph the equation using slope-intercept graphing

The equation \( y = 300x \) is in the slope-intercept form \( y = mx + b \) with \( m = 300 \) and \( b = 0 \). This means the graph is a straight line passing through the origin (0,0) with a slope of 300. Mark points such as \((0, 0)\), \((1, 300)\), and \((2, 600)\) on the graph and draw the line through these points.

Step 6: Use the graph to find the amount of sodium in 3 servings

On the graph, locate the point where \( x = 3 \) servings. Read the corresponding \( y \)-value, which gives the amount of sodium. According to the equation, when \( x = 3 \):\[ y = 300 \times 3 = 900 \text{ mg} \]

Key Concepts

constant of proportionality

In direct variation, the constant of proportionality, denoted as \( k \), describes how one quantity varies directly with another. For instance, in the problem where 6 servings of salad dressing contain 1800 mg of sodium, we establish that the relationship between the amount of dressing, \( x \), and the sodium, \( y \), is given by \( y = kx \). To find \( k \), use the formula:\[ k = \frac{y}{x} \ = \frac{1800 \text{ mg}}{6 \text{ servings}} \ = 300 \frac{\text{mg}}{\text{serving}} \]So, the constant of proportionality here is 300 mg per serving. This tells us that for every serving of salad dressing, there are 300 mg of sodium.

slope-intercept form

The slope-intercept form of a linear equation is helpful for graphing and understanding linear relationships. The general formula is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In our exercise, the equation \( y = 300x \) fits this form perfectly:

• The slope \( m \) = 300
• The y-intercept \( b \) = 0

This indicates a straight line through the origin, with a slope of 300. The slope represents the rate of change of sodium content per serving of salad dressing. Since there's no constant added to \( 300x \), the line starts at (0,0), meaning zero sodium when there are zero servings of dressing.

graphing linear equations

Graphing the equation helps visualize the relationship between servings of salad dressing and sodium content. For the equation \( y = 300x \):

• Start at the origin (0,0) since the y-intercept is 0.
• Next, use the slope, which is 300. This means for every increase in 1 serving of dressing (x-axis), the sodium content (y-axis) increases by 300 mg.

Plot points such as (0, 0), (1, 300), and (2, 600). Draw a straight line through these points, extending in both directions. This line represents the direct variation relationship graphically.

algebraic relationships

Understanding algebraic relationships is key to solving problems involving direct variation. In the given exercise, the relationship between salad dressing servings and sodium content is expressed algebraically as \( y = 300x \).This relationship tells us:*How variables change in relation to each other*. If one variable increases, the other does so proportionally.
Knowing the equation, we can predict sodium content for any number of servings. For example:

• For 16 servings of salad dressing, \( y = 300 \times 16 = 4800 \text{ mg} \).
• To find sodium in 3 servings, \( y = 300 \times 3 = 900 \text{ mg} \).

This formulaic approach makes understanding and computing relationships straightforward. Just plug in the values and solve!