pre-algebra. lesson 8-4 warm-up pre-algebra what is a “direct variation”? how do you find the...
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PRE-ALGEBRA
PRE-ALGEBRA
Lesson 8-4 Warm-Up
PRE-ALGEBRA
What is a “direct variation”?
How do you find the constant, k, of a direct variation given a point on its line?
direct variation (sometimes called a direct proportion) - a linear (line) function (relationship in which y varies, or “changes”, directly with changes in x) that follows the equation: y = kx where the coefficient k is called the “constant of the variation” (a constant is a number that never changes) and k ≠ 0. Note: Since y = 0 when x = 0, all direct variations pass through the origin (0, 0)Examples: y = ¾ x y = -½ x To find the k of a a direct variation, substitute the (x, y) values of a point on its line.Example: Find the constant of variation, k, of the direct variation that contains the point (2, 4)
y = kx Direct Variation Equaton.4 = k(2) Substitute the x and y values of
(2,4) into y = kx.4 = 2k Simplify.2 2 Solve for k by dividing both
sides by 2.2 = k
The constant of variation is 2.
Direct Variation (8-4)
PRE-ALGEBRA
The table of values models a direct variation.
Graph the direct variation. Find the constant of
variation.x
–2
0
2
6
y
–1
0
1
3
Plot the points on a coordinate plane. Connect the points.
Direct VariationLESSON 8-4
Additional Examples
PRE-ALGEBRA
(continued)
y = kx Use the equation for a direct variation.
1 = k • 2 Substitute a point in the table like (2, 1).
The constant of variation is .12
= k Solve for k.12
Use a point to find the constant of variation.
Direct VariationLESSON 8-4
Additional Examples
PRE-ALGEBRA
How can use a direct variation to solve an equation in which there is a proportional relationship between two things?
You can use a direct proportion equation to solve any problem in which there is a proportional relationship between two values (x and y). To do this, find the constant of variation using the given relationship between x and y. Then, you can use substitution to find how the the change in on value affects the other value.
Example: Your weight on Earth’s surface varies directly with your weight on Mars. If a person weighing 120 lb. on Earth weighs about 40 lb. on Mars, how much would a person weighing 180 lb. on Earth weigh on Mars? Step 1: Find the constant of variation.
Let E = weight on Earth Assign variables to weight on Earth andLet M = weight on Mars MarsE = kM Write the equation for a direct variation
using your variables.120 = k • 40 Substitute 120 for E and 40 for M 40 40 Divide each side by 40 to isolate the k. 3 = k Simplify.
Direct Variation (8-4)
PRE-ALGEBRA
Step 2: Find the weight of a 180 pound person on Mars.
E= kM Write the equation for a direct variation using your variables.
180 = 3 · M Substitute 180 for E and 3 for k 3 3 Divide each side by 3 to isolate the M 60 = M Simplify.
A person weighing 180 pounds on Earth weights about 60 pounds on Mars.
Direct Variation (8-4)
PRE-ALGEBRA
The number of miles Joe walks varies directly
with the number of hours he walks. He walks 6 miles in
2 hours. Write a direct variation to find how many hours it
takes Joe to walk 15 miles.
Step 1 Find the constant of variation.
m = kh Write the equation for a direct variation.
6 = k • 2 Substitute 6 for m and 2 for h.
3 = k Divide each side by 2. Simplify.
Direct VariationLESSON 8-4
Additional Examples
PRE-ALGEBRA
Step 2 Write the equation using the value of k.
m = kh Write the equation for a direct variation.
15 = 3h Substitute 15 for m and 3 for k.
5 = h Divide each side by 3. Simplify.
It takes Joe 5 hours to walk 15 miles.
(continued)
Direct VariationLESSON 8-4
Additional Examples
PRE-ALGEBRA
Step 1 Find the constant of variation.
y = kx Write the equation for a direct variation.
– 5 = k(7) Substitute –5 for y and 7 for x.
Step 2 Write the equation using the value of k.
y = kx Write the equation for a direct variation.
k = – Divide each side by 7. Simplify.57
y = – x Replace k with – .57
57
Write an equation for the direct variation that
includes B (7, –5).
Direct VariationLESSON 8-4
Additional Examples
PRE-ALGEBRA
How do you tell whether each “data pair” (x and y) in a table is a direct variation?
Tip: You can write y = kx as k = y / x if you divide both sides by x. If each data pair (x y) equals k (in other words, the ratio of y to x is the same for each x and y pair), then the table represents a direct variation.
Example: Is the following table a direct variation?
No, the ratio of (in other words, the “k”) is not the same for all of the x and y data pairs.
Direct Variation (8-4)
PRE-ALGEBRA
1 –2 = –0.5
–1 2
–2 4
= –0.5
= –0.5
yxx y
–2 1
2 –1
4 –2
2 –1 = –2
21
–4 2
= 2
= –2
yxx y
–1 2
1 2
2 –4
For the data in each table, use the ratio to tell whether y
varies directly with x. If it does, write an equation for the direct
variation.
Yes, the constant of variation is –0.5. The equation is y = –0.5x.
yx
No, the ratio is not the
same for each pair of data.
yx
a. b.
Direct VariationLESSON 8-4
Additional Examples
Direct Variation
PRE-ALGEBRA
Write an equation for a direct variation that includes each point.1. R(3,5) 2. S(6, – 2) 3. T(– 1, –2)
4. Graph the direct variation modeled by the table of values. Find the constant of variation.
5. The number of words Mary types varies directly with the number of minutes she types. She types 90 words in 3 minutes. Write a direct variation to find how many minutes it takes Mary to type 60 words.
y = 2x
w = 30m; 2 minutes
y = x53 y = – x1
3
Direct VariationLESSON 8-4
Lesson Quiz