Using the Distance Formula to Find Coordinates

The problem requires finding the value of $x$ given the coordinates of two points, $(x, 7)$ and $(1, 15)$, and the distance between them, which is 10 units. We can use the distance formula derived from the Pythagorean theorem.

Distance Formula Explanation

The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a Cartesian coordinate system is given by the formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Step-by-Step Calculation

1. Substitute known values into the distance formula:

   Here, $(x_1, y_1) = (x, 7)$, $(x_2, y_2) = (1, 15)$, and $d = 10$.

   $10 = \sqrt{(1 - x)^2 + (15 - 7)^2}$

2. Simplify the equation:

   Calculate the difference in the y-coordinates and square it.

   $10 = \sqrt{(1 - x)^2 + (8)^2}$

   $10 = \sqrt{(1 - x)^2 + 64}$

3. Square both sides to eliminate the square root:

   $10^2 = (1 - x)^2 + 64$

   $100 = (1 - x)^2 + 64$

4. Isolate the term containing $x$:

   Subtract 64 from both sides.

   $(1 - x)^2 = 100 - 64$

   $(1 - x)^2 = 36$

5. Solve for $(1 - x)$ by taking the square root:

   Remember that the square root of 36 can be positive or negative.

   $1 - x = \pm\sqrt{36}$

   $1 - x = 6 \quad \text{or} \quad 1 - x = -6$

6. Solve for $x$ in both cases:
   • Case 1: $1 - x = 6$

     $x = 1 - 6$

     $x = -5$

   • Case 2: $1 - x = -6$

     $x = 1 - (-6)$

     $x = 1 + 6$

     $x = 7$

Conclusion

The possible values of $x$ are 7 and -5.