The problem asks to find the distance from the line $y = \frac{5}{4}x - 4$ to the point $(-1, 5)$.

To find the distance from a point $(x_1, y_1)$ to a line given by $Ax + By + C = 0$, we use the formula:

$d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$

Step 1: Rearrange the line equation

The given equation $y = \frac{5}{4}x - 4$ can be written in standard form: $\frac{5}{4}x - y - 4 = 0$ Multiply through by 4 to clear the fraction: $5x - 4y - 16 = 0$ Here, $A = 5$, $B = -4$, and $C = -16$.

Step 2: Apply the distance formula

Substitute the coordinates of the point $(-1, 5)$ into the formula: $d = \frac{|5(-1) + (-4)(5) - 16|}{\sqrt{5^2 + (-4)^2}}$

Step 3: Simplify the numerator

$d = \frac{| -5 - 20 - 16 |}{\sqrt{25 + 16}}$ $d = \frac{|-41|}{\sqrt{41}}$ $d = \frac{41}{\sqrt{41}}$

Step 4: Rationalize the denominator (if needed)

$d = \sqrt{41}$

Final Answer:

The distance from the point $(-1, 5)$ to the line $y = \frac{5}{4}x - 4$ is: $\sqrt{41}$

Would you like me to elaborate on any step?

5 Related Questions:

1. How is the formula for the distance from a point to a line derived?
2. What happens if the coordinates of the point were different (e.g., $(0, 0)$)?
3. Can you verify this result geometrically or with a graph?
4. What other methods can be used to compute the distance from a point to a line?
5. How do we calculate distances if the line is given in parametric form?

Tip:

Always write the equation of the line in standard form $Ax + By + C = 0$ before applying the distance formula.