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Chapter 1: Problem 113

If \(y\) and \(x\) are related by the linear expression \(y=m x+b\), how will \(y\) change as \(x\) changes if \(m\) is positive? negative? zero?

Short Answer

Expert verified
If the slope \(m\) is positive in the linear expression \(y = mx + b\), y will increase as x increases. If the slope \(m\) is negative, y will decrease as x increases. If the slope \(m\) is zero, y will remain constant regardless of changes in x.

Step by step solution

01

Understand the given linear expression

Given the linear expression, \(y = mx + b\), where: - y and x are the dependent and independent variables, respectively - m is the slope and describes the steepness of the line - b is the y-intercept, representing the point at which the line crosses the y-axis We will analyze how the value of y changes with respect to x depending on the value of the slope (m).
02

Case 1: Positive slope (m > 0)

When the slope m is positive (m > 0), as x increases, y will also increase positively. In this case, the line representing the linear function will slope upwards from left to right on a coordinate plane, indicating a positive correlation between y and x.
03

Case 2: Negative slope (m < 0)

When the slope m is negative (m < 0), as x increases, y will decrease negatively. In this case, the line representing the linear function will slope downwards from left to right on a coordinate plane, indicating a negative correlation between y and x.
04

Case 3: Zero slope (m = 0)

When the slope m is zero (m = 0), the linear expression becomes \(y = b\). In this case, the value of y remains constant and does not change with respect to x. The line representing the linear function will be horizontal on a coordinate plane, indicating no correlation between y and x. In conclusion, depending on the sign of the slope (m), the value of y in the linear expression \(y = mx + b\) will change as follows: - If m is positive, y will increase as x increases - If m is negative, y will decrease as x increases - If m is zero, y will remain constant regardless of changes in x

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