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# Graphing

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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?

Nov 5, 2021

#1
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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 )

Nov 5, 2021