Understanding the Y-Intercept of the Equation 3x - 4y 20

The y-intercept of an equation in the form of a line, such as 3x - 4y 20, is the point where the line crosses the y-axis. This occurs when the value of x is zero. In this article, we will explore how to find the y-intercept of the given equation step-by-step.

Step-by-Step Solution

To find the y-intercept, we start by setting x equal to 0. This is because the y-intercept represents the point where the line crosses the y-axis, and at that point, the x-coordinate is always 0.

Starting with the equation:

3x - 4y 20

Set x 0:

3(0) - 4y 20

This simplifies to:

-4y 20

Next, we solve for y by dividing both sides by -4:

y 20 / -4 -5

Thus, the y-intercept is -5. In coordinate form, the y-intercept is (0, -5).

Conversion to Slope-Intercept Form

To further illustrate the concept, let's convert the equation 3x - 4y 20 into slope-intercept form (y mx c). In this form, m represents the slope and c represents the y-intercept.

Starting with the equation:

3x - 4y 20

First, isolate y:

-4y -3x 20

Divide all terms by -4:

y (-3/4)x (-20/4)

This simplifies to:

y (-3/4)x - 5

Here, the y-intercept (c) is -5, as we found previously.

Multiple Methods for Finding the Y-Intercept

It's worth noting that there are several ways to find the y-intercept of a linear equation. Each method leads to the same result, which is the point where the line crosses the y-axis.

Substituting x 0 directly into the original equation: Isolating y in the slope-intercept form: Using the intercept form of the equation (x/a y/b 1), where (0, b) is the y-intercept.

No matter the method, the y-intercept of the equation 3x - 4y 20 is consistently found to be 5 when x is set to 0. This can be verified by any of the given methods.

Conclusion

The y-intercept of a linear equation is a crucial point that helps in understanding the behavior of the line. As demonstrated, for the equation 3x - 4y 20, the y-intercept is -5. Understanding how to find the y-intercept through various methods not only deepens your knowledge of algebraic equations but also enhances your problem-solving skills in mathematics.