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Proportional relationships

Determine the linear equation that gives the rule for this table.
 x y 0 0 1 10 2 20 3 30

Linear functions are of the form y = mx + b.
First find m. Look at the table and notice that every time the x terms go up by 1, the y terms go up by 10. This means that m is equal to 10.

 x y 0 0 1 10 2 20 3 30

Next find b. Take the equation y = mx + b and plug in the m value (m = 10) and a pair of (x, y) coordinates from the table, such as (0, 0). Then solve for b.

 y
=
 m
 x
+
 b

0  =
10
 ( 0 )
+
 b
Plug in m = 10, x = 0, and y = 0

0  =
0  +
 b
Simplify

0  =
 b
Simplify

Finally, use the m and b values you found (m = 10 and b = 0) to write the equation.

 y
=
 m
 x
+
 b

 y
=
10
 x
+  0
Plug in m = 10 and b = 0

 y
=
10
 x
Simplify

Now check your answer. Plug in each (x, y) pair in the table, and see if the result is a true statement.

Plug in (0, 0).

y  =  10x

0
 ? =
10(0)

0
 ! =
0

Plug in (1, 10).

y  =  10x

10
 ? =
10(1)

10
 ! =
10

Plug in (2, 20).

y  =  10x

20
 ? =
10(2)

20
 ! =
20

Plug in (3, 30).

y  =  10x

30
 ? =
10(3)

30
 ! =
30

Each (x, y) pair from the table resulted in a true statement.

So, the linear equation is y = 10x.

During a thunderstorm you see the lightning before you hear the thunder because light travels much faster than sound. The distance between you and the storm varies directly as the time interval between the lightning and the thunder. Suppose that the thunder from a storm 5400 ft away takes 5 s to reach you. Determine the constant of proportionality, and write the equation for the variation.

Let d be the distance from you to the storm, and let t be the length of the time interval. We are given that d varies directly as t, so

d=kt

where k is a constant. To find k, we use the fact taht t=5 when d=5400.
Substituting these values in the equation, we get

 5400 = k(5) Substitute k = $\frac{5400}{5}$ = 1080 Solve for k

Substituting this value of k in the equation for d, we obtain

d= 1080t

as the equation for d as a function of t.