*25 points! * state in detailed terms on how to solve a three variable system, complete with an example.

Pick 2 pairs of equations then use addition and subtraction to eliminate the same variable from both pairs of equations then it is left with 2 variables 
Pick two pairs
4x - 3y + z = - 102x + y + 3z = 0
eliminate the same variable from each system
4x - 3y + z = - 10 
2x + y + 3z = 0 

4x - 3y + z = - 10 
-4x - 2y - 6z = 0 

-5y - 5z = - 10 

2x + y + 3z = 0 
- x + 2y - 5z = 17 

2x + y + 3z = 0 
-2x + 4y - 10z = 34 

5y - 7z = 34
Solve the system of the two new equations:
-5y - 5z = - 10 
5y - 7z = 34 

-12z = 24 
which is , z = - 2 

-5y - 5(- 2) = - 10 
-5y = - 20 
wich is , y = 4
substitute into one of the original equations
- x + 2y - 5z = 17 
- x + 2(4) - 5(- 2) = 17 
- x + 18 = 17 
- x = - 1 
x = 1 
which is , (x, y, z) = (1, 4, - 2) 
Does 2(1) + 4 + 3(- 2) = 0 ? Yes

Intersecting secant-tangent theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.so to break it down: 1. pq acts as the tangent and ps as the secant with an intersection at p.sr = 21rp = 3x + 3pq = 4x + 4following to equation: tangent squared = the two segments multiplied by each otherso (4x + 4) ^ 2 = (3x + 3) • (21) simplifies to 16x^2 - 31x - 47 = 0 or (16x - 47)(x + 1) x = -1 or x = 47/162. pq acts as the tangent and ps as the secant with an intersection at p (again! )sr = 16rp = xpq = 15(15) ^ 2 = 16 • x225 = 16xx = 225/163. the interesting chords are proportional to each other so a ratio is possible to set up: ap dp 10 3 + x = = —— = pc pb 8 xcross multiple to get 10x = (3 + x)(8)x = 12