Please someone help me. I’m struggling so work this out.

First of all, look at what the values are in the tables. You are looking to see if the x-values are increasing by 1. Except in problems 5 and 6, that is the case. In those problems, the last x-value is not in sequence. (Not a problem.)

Second, look at the differences between y-values. When the change in x is the same, you want the change in y to be the same. If it s, the rule is a linear rule. (That is the case for all of the problems here.)

Since each rule is a linear rule, it can be written in any of several forms. Your final answer may want to be in "slope-intercept" form, which look__s like ...

y = mx + b . . . . . . slope-intercept form

To get there, you may want to start with "point-slope" form, which can look like ...

y = (y2 -y1)(x -x1) +y1 . . . . . . when x2=x1 +1

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Using this second form on these problems gives ...

(1) y = (11 -5)(x -1) +5

y = 6x -1 . . . . . . . . . . . simplified to slope-intercept form

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(2) y = (15 -11)(x -2) +11

y = 4x +3 . . . . . . . . . . . simplified to slope-intercept form

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(3) y = (6 -3)(x -3) +3

y = 3x -6 . . . . . . . . . . . simplified to slope-intercept form

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(4) y = (19 -10)(x -2) +10

y = 9x -8 . . . . . . . . . . . simplified to slope-intercept form

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(5) y = (14 -16)(x -(-3)) +16

y = -2x +10 . . . . . . . . . . . simplified to slope-intercept form

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(6) This one is a bit tricky, since the x-values in the table are decreasing, not increasing. One way to deal with that is to reverse the first two table entries, so the first one is (x, y) = (-3, -8) and the second one is (-2, -5). Then our formula gives ...

y = (-5 -(-8))(x -(-3)) -8

y = 3x +1 . . . . . . . . . . . simplified to slope-intercept form

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Comment on the "point-slope" formula

For arbitrary (x, y) pairs, the "two point" form of the equation of a line is ...

y = (y2 -y1)/(x2 -x1)(x -x1) +y1

In this set of problems, (x2 -x1) = 1 (for all but the last), so we shortened the form to the one we used above:

y = (y2 -y1)(x -x1) +y1

When the change in x is 1, as for most of these problems, then the change in y is the slope of the line. That is why we called the reduced form here "point-slope" form. It made use of the first point (x1, y1) and the slope (y2 -y1).

This reduced "point-slope" form cannot be used except under the special circumstances that exist for this problem set. (In problem 6, we had to rearrange the problem to create the right circumstances.)