Understanding the Gradient and Y-Intercept of a Line: A Step-by-Step Guide

In the context of linear equations, it is essential to understand the gradient (slope) and y-intercept. This guide will walk you through the process of finding these values for a specific example, y 3x - 7.

What is the Gradient (Slope)?

The gradient, also known as the slope, of a line is given by the coefficient of x in the equation of a line. It represents the rate of change of y with respect to x. For the equation y 3x - 7, the gradient is 3. This means that for every unit increase in x, y will increase by 3 units.

What is the Y-Intercept?

The y-intercept is the point where the line crosses the y-axis. When x 0, the equation provides the value of y. For the equation y 3x - 7, substituting x 0 gives:

y 3(0) - 7 -7

Therefore, the y-intercept is -7, which corresponds to the point (0, -7).

General Steps to Find Gradient and Y-Intercept

To find the gradient and y-intercept for any linear equation of the form y mx b:

Identify the gradient (m): This is the coefficient of x in the equation. Identify the y-intercept (b): This is the constant term in the equation, or the value of y when x 0.

For the equation y 3x - 7:

Gradient (m) 3 Y-intercept (b) -7

Additional Context and Examples

For a more general equation of a line in the form:

ax by c, the gradient (m) is given by m -a/b, and the y-intercept is c/b. y - k m(x - h), the gradient (m) remains the same, and the y-intercept is given by y-intercept k - mh.

Conclusion

Understanding the gradient and y-intercept is crucial in analyzing and graphing linear equations. By following the steps outlined in this guide, you can easily determine these key features for any given line equation. Whether you are working with specific values like y 3x - 7 or more general equations, the techniques remain the same.

Key Takeaways

Gradient (slope) is the coefficient of x in the equation y mx b. Y-intercept is the value of y when x 0, or the constant term in the equation. For a line equation ax by c, the gradient is m -a/b, and the y-intercept is c/b.