For every value of a, the graph of each of these three equations is a line:
ax + y = 1
x + ay = 1
x + y = 9
For what value of a do all three of these lines intersect at the same point?
+23245 Solve the first equation for y: ax + y = 1 ---> y = 1 - ax
Solve the second equation for y: x + ay = 1 ---> y = (1 - x) / a
Solve the third equation for y: x + y = 9 ---> y = 9 - x
Combine the first and third equations: 1 - ax = 9 - x
-8 = ax - x
-8 = (a - 1)x ---> x = -8 / (a - 1)
---> x = 8 / (1 - a)
Combine the second and third equations: (1 - x) / a = 9 - x
1 - x = 9a - ax
1 - 9a = x - ax
1 - 9a = (1 - a)x ---> x = (1 - 9a) / (1 - a)
Combine the two resulting equations: 8 / (1 - a) = (1 - 9a) / (1 - a)
Multiply both sides by (1 - a): 8 = 1 - 9a
7 = -9a
a = -7/9
And, the point where they intersect is: ( 4.5, 4.5 )
