Problem 5 - Applied Calculus

Question

Given the equation \(y=5 x-12,\) how will \(y\) change if \(x\) : a. Increases by 3 units? b. Decreases by 2 units?

Step-by-Step Solution

Answer

a. \( y \) increases by 15 units. b. \( y \) decreases by 10 units.

Step 1: Understand the relationship

The equation given is a linear equation of the form \( y = 5x - 12 \). This equation shows that \( y \) is dependent on \( x \) and that for every unit change in \( x \), \( y \) changes by \( 5 \) times that amount because the coefficient of \( x \) is \( 5 \).

Step 2: Calculate change in y when x increases by 3 units

When \( x \) increases by \( 3 \) units, the change in \( x \) is \( +3 \). Since \( y = 5x - 12 \), the change in \( y \) is \( 5 \times 3 = 15 \). Therefore, \( y \) increases by \( 15 \) units when \( x \) increases by \( 3 \) units.

Step 3: Calculate change in y when x decreases by 2 units

When \( x \) decreases by \( 2 \) units, the change in \( x \) is \( -2 \). Using the same pattern, the change in \( y \) is \( 5 \times (-2) = -10 \). Therefore, \( y \) decreases by \( 10 \) units when \( x \) decreases by \( 2 \) units.

Key Concepts

Change in Variables

In any linear equation like the one given, changes in the variables can give us insights into how the values are interconnected. Here, the equation is represented as \( y = 5x - 12 \). This suggests that \( y \) depends directly on \( x \).
When we adjust the value of \( x \) by a set amount, say increase by 3 or decrease by 2, we can predict the change in \( y \) using the equation base.

• Increasing \( x \) by 3 results in \( y \) increasing because we're multiplying \( 3 \) by the coefficient of \( x \), which is \( 5 \).
• On the other hand, decreasing \( x \) by 2 makes \( y \) decline, as we're multiplying \( -2 \) by \( 5 \).

Understanding these changes helps us predict outcomes, ensuring we're prepared and informed about the relationships in the equation.

Equation Analysis

Breaking down the equation \( y = 5x - 12 \) can reveal how each element affects the outcome. This equation is linear, meaning it forms a straight line when graphed.

• Coefficient of \( x \): In this exercise, the coefficient is \( 5 \), meaning for every unit increase or decrease in \( x \), \( y \) changes proportionally by \( 5 \) times that amount.
• Constant term \(-12\): This remains fixed and doesn't change with \( x \). It shifts the entire line up or down on the graph, depending on its value.

By understanding each part of the equation, you can determine how and why \( y \) changes when \( x \) does. This is crucial for problem-solving and ensuring clarity in results.

Dependent and Independent Variables

In mathematical equations like \( y = 5x - 12 \), variables play distinct roles. The terms 'dependent' and 'independent' are pivotal in understanding these relationships.

• Independent Variable \( x \): This is the variable you have control over. You can choose its value, making it the driver of the equation.
• Dependent Variable \( y \): This variable's value depends on \( x \). As \( x \) changes, \( y \) responds according to the given equation.

Understanding these terms allows for logical deductions about outcomes from changes in \( x \). It's a foundational skill in ensuring comprehension of how equations model real-world scenarios.