Understanding the Y-Intercept of the Line: (frac{1}{4}x - y -1)

The y-intercept of a line 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 line given by the equation (frac{1}{4}x - y -1).

Step 1: Rewriting the Equation

First, let's write the equation in the standard form (y mx b), where (m) is the slope and (b) is the y-intercept.

(frac{1}{4}x - y -1)

Subtract (frac{1}{4}x) from both sides to isolate (y):

( -y -1 - frac{1}{4}x )

Now, multiply through by -1 to solve for (y):

( y frac{1}{4}x 1 )

Step 2: Finding the Y-Intercept

The y-intercept occurs when (x 0). So, we substitute (x 0) into the equation:

( y frac{1}{4}(0) 1 1)

Therefore, the y-intercept is (1). This means the line crosses the y-axis at the point ((0, 1)).

Alternative Method: Direct Substitution

Alternatively, we can directly substitute (x 0) into the original equation to find the y-intercept:

(frac{1}{4}(0) - y -1)

This simplifies to:

( -y -1 )

Multiplying both sides by -1:

( y 1 )

The y-intercept is still (1), confirming our earlier result.

Conclusion

The y-intercept of the line defined by the equation (frac{1}{4}x - y -1) is (1). This is the value of (y) when (x 0), and the line crosses the y-axis at the point ((0, 1)).

Understanding how to find the y-intercept is crucial in interpreting linear equations and graphing lines.