Graphs: 14, 16, and 17 are graphs of proportional relationships. The constants of proportionality are 3/2, -1/4, and 1, respectively.
Missing values: 18: 12; 19: 6; 20: 21; 21: -4; 22: -5; 23: 40.
Step-by-step explanation:
Explanation for Graphs
The graph of a proportional relation is always a straight line through the origin. The graph of 15) is not such a graph, so is not the graph of a proportional relation.
The constant of proportionality is the slope of the line: the ratio of vertical change to horizontal change. In each of these graphs, points are marked so it is easy to count the squares between marked points to determine the amount of change. (One of the marked points in each case is the origin.)
14) The graph goes up 3 for 2 squares to the right, so the slope and constant of proportionality are 3/2.
16) The graph goes down 1 square for 4 squares to the right, so the slope and constant of proportionality are -1/4.
17) The graph goes up 3 squares for 3 squares to the right, so the slope and constant of proportionality are 3/3 = 1.
Explanation for Missing Values
When 3 values are given and you're asked to find the 4th in a proportion, there are several ways you can do it. Here's one that may be easy to remember, especially if you write it down for easy reference when you need it.
Let's call the given values "a", "b", and "c". They can be given in ordered pairs, such as (x, y) = (a, b) = (2, -4), and a missing value from an ordered pair, such as (c, _) = (-6, y). (These are the numbers from problem 18.)
In this arrangement, the "_" is the second value of the second ordered pair, so corresponds to "b", the second value of the first ordered pair. The value "c" is the other half of the ordered pair with a value missing, so it, too, can be said to correspond to the "_".
The solution is the product of these two corresponding values, divided by the remaining given value. That is, for ...
... (a, b) = (c, _)
the unknown value is
... _ = bc/a
___
If the relation is written with the first value missing, the same thing is true: the solution is the product of corresponding values divided by the remaining given value.
... (a, b) = (_, c)
... _ = ac/b
___
This still holds when the pairs are on the other side of the equal sign.
For (c, _) = (a, b), the solution is _ = bc/aFor (_, c) = (a, b), the solution is _ = ac/b
18) y = (-6)(-4)/2 = 12
19) x = (4)(24)/16 = 6
20) y = (12)(7)/4 = 21
21) x = (-16)(6)/24 = -4
22) x = (3)(30)/-18 = -5
23) x = (32)(100)/80 = 40
More Formally ...
In more formal terms, the proportional relation can be written as
... b/a = _/c . . . . for (a, b) = (c, _)
Multiplying both sides of this equation by c gives ...
... bc/a = c_/c
Simplifying gives
... bc/a = _
When the missing value is the other one in the ordered pair, we can still write the proportion with the missing value in the numerator, then solve by multiplying the equation by the denominator under the missing value.
... a/b = _/c . . . . for (a, b) = (_, c)
... _ = ac/b
