What is the rate of change for the linear equation y = -4x + 9?
The correct answer and explanation is:
Correct Answer:
The rate of change for the linear equation y = -4x + 9 is -4.
Explanation (300 words):
The rate of change in a linear equation refers to how much the dependent variable (usually y) changes in response to a one-unit change in the independent variable (usually x). This concept is fundamental in understanding how variables are related in algebra, particularly in the study of linear functions.
The general form of a linear equation is: y=mx+by = mx + b
In this equation:
- m represents the slope, which is also the rate of change.
- b represents the y-intercept, the value of y when x = 0.
Now, let’s analyze the given equation: y=−4x+9y = -4x + 9
Here, the coefficient of x is -4. This coefficient is the slope of the line, which tells us the rate at which y changes for each one-unit increase in x.
What does a rate of change of -4 mean?
It means that for every increase of 1 in x, the value of y decreases by 4. For example:
- If x = 0, then y = 9
- If x = 1, then y = -4(1) + 9 = 5
- If x = 2, then y = -4(2) + 9 = 1
As you can see, each time x increases by 1, y decreases by 4. This consistent change is what makes the equation linear.
Why is understanding rate of change important?
In real-world contexts, the rate of change helps us understand things like speed (distance over time), cost (price per item), and trends (such as temperature or population change over time). In this case, a negative rate indicates a decreasing trend.
Thus, the rate of change for the equation y = -4x + 9 is -4.