Math question...equations...graphs etc..?

Question

A, B, C, D, E, and F are all integers and no two are the same integer.
(a) Find values for A, B, C, D, E, and F so that the linear equations Ax + By = C and
Dx + Ey = F have the same solution (−3,4) .
(b) Sketch a graph to represent your two equations.
(c) Is your set of values for A, B, C, D, E, and F unique? Explain how you know.

Answer 1

1a) Solve each in terms of y:
By = C - Ax
y = C/B - A/B(x)
y = -A/B(x) + C/B

And it must pass through the point (-3, 4)
4 = 3A/B + C/B
4B = 3A + C
Let's arbitrarily pick values that satisfy this equation:
A = 9, B = 7, C = 1
y = -9/7(x) + 1/7

Second line:
y = -D/E(x) + F/E
4 = 3D/E + F/E
4E = 3D + F
Pick arbitrary values that satisfy this equation:
D = 6, E = 5, F = 2
y = -6/5(x) + 2/5

1b) Sketch these as a graph. They will be two lines passing through the same point (-3, 4)

1c) Since there can be an infinite number of lines passing through this point (-3, 4), our solution cannot be unique.

Just to prove that, here's another solution for A,B,C:
4B = 3A + C
A = 11, B = 12, C = 15

And another for D,E,F:
4E = 3D + F
D = 20, E = 16, F = 4

Answer 2

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