Slope $=-\frac{5}{3}$

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31-10-2026
Mathematics

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Slope $=-\frac{5}{3}$

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-10-31T16:50:42-04:00

Identify that the slope $m = -\frac{5}{3}$ indicates a vertical decrease of 5 units for every 3 units of horizontal increase.
Utilize the slope-intercept form $y = mx + b$ to express the relationship.
Substitute the given slope into the formula to obtain the general linear equation $y = -\frac{5}{3}x + b$.
Conclude that the specific case passing through the origin is $\boxed{y = -\frac{5}{3}x}$.

Explanation

Analysis of the Slope Condition

The slope, denoted by $m$, represents the rate of change of a linear relationship. Given a slope of $m = -\frac{5}{3}$, this indicates that for every 3 units of horizontal movement to the right, the line moves 5 units vertically downward. This value characterizes the steepness and direction of a line in the coordinate plane. Since no specific point or y-intercept is provided, we can define the general linear relationship as $y = mx + b$, where $b$ is the y-intercept. Substituting the given slope, the general equation for such a line is $y = -\frac{5}{3}x + b$.

Formulating the Linear Equation

A linear equation of the form $y = -\frac{5}{3}x + b$ represents a set of all points $(x, y)$ that satisfy the condition of having a constant rate of change of $-\frac{5}{3}$. If we assume the simplest case where the line passes through the origin $(0, 0)$, the y-intercept $b$ is 0, and the equation simplifies to $y = -\frac{5}{3}x$. Any other specific value for $b$ would translate this line vertically along the coordinate plane.

Conclusion and Final Equation

In summary, the slope $-\frac{5}{3}$ dictates the behavior of the linear function. For any change $\Delta x$, the change in $y$ is $\Delta y = -\frac{5}{3} \Delta x$. This confirms the downward trend of the line as it moves from left to right. The general equation describing this relationship is $y = -\frac{5}{3}x + b$, where $b$ is any real number constant representing the y-intercept. If we define the line passing through the origin as the reference, we have $\boxed{y = -\frac{5}{3}x}$.

Examples

Consider a situation where a hiker is descending a mountain path. If the hiker moves forward 3 meters horizontally while dropping 5 meters in elevation, the slope of the path is $\frac{\text{rise}}{\text{run}} = \frac{-5}{3} = -1.67$. Understanding this slope allows engineers and hikers to calculate the total change in elevation over a given horizontal distance, helping in planning safe trails and estimating travel time based on the steepness of the terrain.