The linear system that has the solution x = 12 and y = 17 is?

Intersection point (P): (x, y) = (12, 17)

Equation 1: a(x) + b(y) = constant

Equation 2: c(x) + d(y) = different constant.

The values of a,b,c,d will be different so that the lines are different. Now I will update the system by inputting the (x, y) coordinate of intersection:

Equation 1: 12(a) + 17(b) = constant

Equation 2: 12(c) + 17(d) = different constant.
Select different values for a,b,c,d. I will let a = 1, b = 2, c = 3, d = 4. The point is to find the value of the constants.

Equation 1: 12(1) + 17(2) = 46

Equation 2: 12(3) + 17(4) = 104
Now act like x & y are unknown and input the values of a,b,c,d, constants to the system of equations:

Equation 1: x + 2(y) = 46

Equation 2: 3x + 4(y) = 104

We are done, and there are other solutions since their are infinitely many lines which I could get to pass through this point.

x + y = 29 ; x - y = - 5....3x - 2y = 2 & 4x - 3y = - 3

x+y=29
y=x+5

There are infinitely many linear systems that have solution x = 12, y = 17
2x + y = 41
2x - y = 7
x + 2y = 46
3x - 2y = 2
5x - 3y = 9
x + y = 29

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