Find a point on the x-axis which is equidistant from the points (7, 6) and (3, 4).

Steps to Solve

1. Identify the points and the x-axis point We need to find a point $(x, 0)$ on the x-axis that is equidistant from points $(7, 6)$ and $(3, 4)$.

2. Use the distance formula The distance formula is given by: $$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

3. Set up the distance equations Set the distance from $(x, 0)$ to $(7, 6)$ equal to the distance from $(x, 0)$ to $(3, 4)$: $$ \sqrt{(x - 7)^2 + (0 - 6)^2} = \sqrt{(x - 3)^2 + (0 - 4)^2} $$

4. Square both sides to eliminate the square roots Squaring both sides gives: $$ (x - 7)^2 + 36 = (x - 3)^2 + 16 $$

5. Expand and simplify the equation Expanding both sides: $$ (x^2 - 14x + 49) + 36 = (x^2 - 6x + 9) + 16 $$

This simplifies to: $$ x^2 - 14x + 85 = x^2 - 6x + 25 $$

6. Combine like terms and solve for x Subtract $x^2$ from both sides: $$ -14x + 85 = -6x + 25 $$

Rearranging gives: $$ -14x + 6x = 25 - 85 $$ $$ -8x = -60 $$

Now, solve for $x$: $$ x = \frac{-60}{-8} = 7.5 $$

The point on the x-axis is $(7.5, 0)$.