Q:

What is a linear rule, and how does one write an equation with it?

A:

The linear rule is the idea that any point in a function falls on the same line, and writing an equation with the linear rule only requires knowledge of the function and y-intercept of the function. If two points on the line are known, it's possible to find this information.

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Linear functions are written in the slope-intercept form of f(x) = mx + b, where m is the slope and b is the intercept. If the function is already in this form, substitute a value for x into the equation and solve it for y, then do the same for another value of x. Then, draw a line connecting the two points.

If two points are provided, determine the change in y and divide it by the change in x. For example, if the two points are (3, 2) and (5, 3), y increases by 1 and x increases by 2. This gives the resulting equation a value of 1/2 for m.

Now, it's possible to find the value for b by substituting one point into the equation. For the first point, y is 2. The equation is 2 = 1/2(3) + b, or 2 = 1.5 + b. Solving yields b as 0.5. Checking the other point for validity yields 3 = 1/2(5) + 0.5, which is true.

Therefore, the equation for the function is f(x) = 1/2x + 0.5.

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