To find the distance from the point $(2, 1)$ to the line $4x - 3y + 5 = 0$, we use the formula for the distance from a point to a line in the plane. The formula is given by:

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

where $(x_1, y_1)$ is the point and $Ax + By + C = 0$ is the line.

For our specific problem:

• The point is $(2, 1)$.
• The line equation is $4x - 3y + 5 = 0$, so $A = 4$, $B = -3$, and $C = 5$.

Substitute these values into the formula:

$d = \frac{|4(2) - 3(1) + 5|}{\sqrt{4^2 + (-3)^2}}$

Calculate the numerator:

$|4(2) - 3(1) + 5| = |8 - 3 + 5| = |10| = 10$

Calculate the denominator:

$\sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

Thus, the distance $d$ is:

$d = \frac{10}{5} = 2$

Therefore, the distance from the point $(2, 1)$ to the line $4x - 3y + 5 = 0$ is $2$.

Would you like any more details or have any questions?

Here are some related questions you might find interesting:

1. How do you derive the formula for the distance from a point to a line?
2. What is the geometric interpretation of the distance from a point to a line?
3. How would the distance change if the line equation was different?
4. How do you find the perpendicular distance from a point to a plane in 3D space?
5. Can you find the distance between two parallel lines?
6. How does this method change for non-linear equations?
7. What are some applications of point-to-line distance calculations in real life?
8. How would you find the shortest distance between two skew lines in space?

Tip: Always double-check your values when substituting into formulas to avoid simple arithmetic errors.