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The general form of a linear function is:
f(x) = ax + b
If the equation can be placed into this form, then it is linear. Otherwise, it is not.
2x^2 is a quadratic (order 2) polynomial. Thus, 2x^2-1 is not linear.
A linear function has the shape of a line. Higher order polynomials have curves. This is another way to tell that the equation is not linear.
Linear equations are special because it is easy to interpolate the exact answer if you know two points on a line, and want to find a third point. Because linear equations do not have variables that are products of one another, they are often easy to solve. This makes them very useful in math and science.
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