Understanding the Slope of the Line y -5x - 2 in Slope-Intercept Form
The equation of a line in slope-intercept form is given by y mx b, where m represents the slope and b represents the y-intercept. This form is particularly useful for quickly identifying the slope and y-intercept of a line, without needing to rearrange the equation.
In the equation y -5x - 2, the slope can be directly identified as -5. This is because the coefficient of x is -5, which corresponds to m in the slope-intercept form. Therefore, the slope of the line is -5 .
Slope-Intercept Form and the Given Line
For the equation of line L1, y -5x - 2, this equation is already in slope-intercept form. We can see that:
Slope (m): -5 y-intercept (b): -2The slope intercept provides a clear visual understanding of the line's behavior and its position on the coordinate plane. Here is a quick summary:
For line L1, Slope m -5 and
.
Gradient and Y-Intercept
The gradient (also known as the slope) of a line is often denoted as m and can be calculated as the change in y over the change in x, denoted as dy/dx. In the given equation y -5x - 2, the gradient is clearly -5.
The y-intercept is the value of y when x 0. This can be found by setting x 0 in the equation:
When x 0, y -2. Thus, the y-intercept is -2.
Additional Insights
To further reinforce the concept, consider comparing the given equation with a different form. For example, if we compare the equation y -5x - 2 with the general form y 5x - 2:
Slope (m) of y 5x - 2: 5 y-intercept (b) of y 5x - 2: -2This comparison highlights the distinct effects of positive and negative slopes on the direction of the line. A negative slope, like in y -5x - 2, indicates that the line slopes downwards as we move from left to right.
Conclusion
The slope-intercept form allows us to easily extract the slope and y-intercept of a line. For the equation y -5x - 2, the slope is clearly -5 and the y-intercept is -2. Understanding these components is crucial for analyzing and graphing linear equations. Additionally, tools like CameraMath can be helpful for solving and visualizing more complex mathematical problems.
Understanding the slope and y-intercept in a line's equation is a fundamental concept in algebra and is widely applicable in fields such as calculus, physics, and engineering.