Explanation:
a linear function must follow the structure
y = ax + b
we use the given data point numbers for x and y to solve for the 2 variables a and b.
we use 2 data points to get a solution for the 2 variables. if the third data point then fits into the same equation, then all 3 points are on a line, and the function is therefore linear. if not, then not.
-8 2/5 = a×-7 1/2 + b
-6 = a×-1 1/2 + b
let's subtract the second from the first equation to eliminate b and solve for a :
-8 2/5 = a×-7 1/2 + b
- -6 = a×-1 1/2 + b
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-2 2/5 = a×6 0
6a = -2 2/5 = (-2×5 - 2)/5 = -12/5
a = -12/5 / 6 = -2/5
and now we use one of the 2 original equations with the newly found information to solve for b :
e.g.
-6 = -2/5 × -1 1/2 + b = -2/5 × -3/2 + b = 6/10 + b = 3/5 + b
-6 - 3/5 = b
(-6×5 - 3)/5 = b
-33/5 = b
so, our proposed linear function is
y = -2x/5 - 33/5
let's see, if the third point is on the line (then the equality is true for its coordinates) or not :
-2 = -2/5 × 8 1/2 - 33/5 = -2/5 × (8×2 + 1)/2 - 33/5
-2 = -2/5 × 17/2 - 33/5 = -34/10 - 33/5 = -34/10 - 66/10
-2 = -100/10 = -10
that is not true (-2 is not equal to -10), so the third point is not on the line between the first 2 points.
that means the function is nonlinear.