Key Concepts and Formulas

• Solving a System of Linear Equations: Finding the values of variables that satisfy all equations in the system simultaneously. Methods include substitution, elimination, and matrix methods.
• Equation of a Line Passing Through Two Points: Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the equation of the line passing through them is given by $\frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1}$.
• Perpendicular Distance from a Point to a Line: The distance $d$ of a point $P(x_1, y_1)$ from a line $Ax + By + C = 0$ is given by $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$.

Step-by-Step Solution

Step 1: Finding Point A

Point A is the intersection of the lines $3x + 2y = 14$ and $5x - y = 6$. To find A, we solve this system of linear equations.

1. Multiply the second equation by 2 to eliminate $y$: $2(5x - y) = 2(6) \Rightarrow 10x - 2y = 12$
2. Add the modified second equation to the first equation: $(3x + 2y) + (10x - 2y) = 14 + 12$ $13x = 26$
3. Solve for $x$: $x = \frac{26}{13} = 2$
4. Substitute $x = 2$ into the second equation $5x - y = 6$: $5(2) - y = 6$ $10 - y = 6$ $y = 4$

Therefore, the coordinates of point A are $(2, 4)$.

Step 2: Finding Point B

Point B is the intersection of the lines $4x + 3y = 8$ and $6x + y = 5$. To find B, we solve this system of linear equations.

1. Multiply the second equation by 3 to eliminate $y$: $3(6x + y) = 3(5) \Rightarrow 18x + 3y = 15$
2. Subtract the first equation from the modified second equation: $(18x + 3y) - (4x + 3y) = 15 - 8$ $14x = 7$
3. Solve for $x$: $x = \frac{7}{14} = \frac{1}{2}$
4. Substitute $x = \frac{1}{2}$ into the second equation $6x + y = 5$: $6\left(\frac{1}{2}\right) + y = 5$ $3 + y = 5$ $y = 2$

Therefore, the coordinates of point B are $\left(\frac{1}{2}, 2\right)$.

Step 3: Finding the Equation of Line AB

We have A$(2, 4)$ and B$\left(\frac{1}{2}, 2\right)$. We use the two-point form to find the equation of the line AB.

1. The equation of the line is: $\frac{y - 4}{x - 2} = \frac{2 - 4}{\frac{1}{2} - 2}$ $\frac{y - 4}{x - 2} = \frac{-2}{-\frac{3}{2}}$ $\frac{y - 4}{x - 2} = \frac{4}{3}$
2. Cross-multiply: $3(y - 4) = 4(x - 2)$ $3y - 12 = 4x - 8$
3. Rearrange to the general form $Ax + By + C = 0$: $4x - 3y + 4 = 0$

Step 4: Finding the Distance from Point P to Line AB

We want to find the distance from $P(5, -2)$ to the line $4x - 3y + 4 = 0$. We use the perpendicular distance formula:

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

1. Substitute $x_1 = 5$, $y_1 = -2$, $A = 4$, $B = -3$, and $C = 4$: $d = \frac{|4(5) - 3(-2) + 4|}{\sqrt{4^2 + (-3)^2}}$ $d = \frac{|20 + 6 + 4|}{\sqrt{16 + 9}}$ $d = \frac{|30|}{\sqrt{25}}$ $d = \frac{30}{5}$ $d = 6$

Common Mistakes & Tips

• Be careful when solving systems of linear equations. A small arithmetic error can lead to an incorrect point of intersection. Double-check your calculations.
• Ensure the equation of the line is in the general form $Ax + By + C = 0$ before using the perpendicular distance formula.
• When calculating the slope using two points, make sure to subtract the coordinates in the same order in both the numerator and the denominator.

Summary

We found the coordinates of points A and B by solving the given systems of linear equations. Then, we determined the equation of line AB using the two-point form. Finally, we calculated the perpendicular distance from point P to line AB using the perpendicular distance formula. The distance is 6.

Final Answer

The final answer is \boxed{6}, which corresponds to option (D).