Algerbra I

Question

Give one ordered pair that is a solution to the equaton A(-9, 10) and B(9, 2). Mathematically, explain why this point is a solution

 

Answer 1

y-10=a(x+9), ax-y+9a+10=0 has the solution A(-9,10);

y-2=b(x-9), bx-y-9b+2=0 has the solution B(9,2), where a and b are unknown constants (slopes), and a,b≠0, a≠b.

There are multiple lines passing through A and multiple lines passing through B. We don't know their slopes (an and b), but the point to be found is the intersection of those lines. Therefore the intersection point depends on the unknown slopes.

So, subtracting one equation from the other:

(a-b)x+9(a+b)+8=0, (a-b)x=-9(a+b)-8, x=(9a+9b+8)/(b-a).

y=a(x+9)+10=a(9+(9a+9b+8)/(b-a))=a(18b+8)/(b-a),

y=2a(9b+4)/(b-a).

So the ordered pair which satisfies both equations is:

((9a+9b+8)/(b-a),2a(9b+4)/(b-a)).

EXAMPLES

If b=a+1 then this pair becomes: (9a+9a+9+8,2a(9a+9+4)),

(18a+17,18a²+26a).

If a=1, b=2 and the ordered pair is (35,44).