Answered

As the $x$-values increase by 1, the $y$-values increase by 15. Which equation shows this?


A$y = 15 + x$
B$y = 15 - x$
C$y = 15x$

Verified Answer :

  • Identify the rate of change \(m = \frac{\Delta y}{\Delta x} = \frac{15}{1} = 15\).
  • Examine each given equation to see which has a slope of \(15\).
  • Recognize that \(y = 15x\) correctly models the linear growth where every unit increase in \(x\) leads to a \(15\)-unit increase in \(y\).
  • Final answer: \(\boxed{c}\)

Explanation

  1. Analyzing the Relationship Between Variables

We are analyzing a linear relationship between two variables, \(x\) and \(y\). A linear relationship is generally expressed in the form \(y = mx + b\), where \(m\) represents the slope (rate of change) and \(b\) is the \(y\)-intercept. We are given that when the \(x\)-values increase by \(1\) (i.e., \(\Delta x = 1\)), the corresponding \(y\)-values increase by \(15\) (i.e., \(\Delta y = 15\)).

  1. Calculating the Slope and Identifying the Linear Form

The rate of change, or slope \(m\), is defined as the change in \(y\) divided by the change in \(x\):
\[m = \frac{\Delta y}{\Delta x} = \frac{15}{1} = 15.\]
A linear equation with a slope of \(15\) must take the form \(y = 15x + b\). Among the provided choices, we need to find which equation corresponds to this rate of change.

  1. Evaluating the Given Options

Now, let's examine the three provided options:

A\(y = x + 15\): This is a linear equation with a slope \(m = 1\). As \(x\) increases by \(1\), \(y\) also increases by \(1\).
B\(y = 15 - x\): This is a linear equation with a slope \(m = -1\). As \(x\) increases by \(1\), \(y\) decreases by \(1\).
C\(y = 15x\): This is a linear equation with a slope \(m = 15\). As \(x\) increases by \(1\), \(y\) increases by \(15 \times 1 = 15\).

Clearly, option (c) matches our calculated slope.

  1. Conclusion

Based on our analysis, the equation \(y = 15x\) correctly models the described relationship where an increase of \(1\) in \(x\) results in an increase of \(15\) in \(y\). Thus, the correct choice is (c). \(\boxed{c}\)

Examples

In real-life scenarios, linear relationships are everywhere. For example, if you are saving money in a piggy bank and you add \$15 each day, your total savings @@MATH_TOKEN_49@@ after @@MATH_TOKEN_50@@ days would follow the equation @@MATH_TOKEN_51@@. This shows how your total money grows at a steady rate of \$15 per day, which is a perfect example of a linear model with a constant rate of change. Understanding this helps in financial planning, predicting growth, and resource management.

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