LESSON 96 JOINT AND COMBINED VARIATION MORE ON
LESSON 96: JOINT AND COMBINED VARIATION, MORE ON IRRATIONAL ROOTS
To review the concept of direct variation, we recall that if the number of peaches varies directly as the number of apples, either of the following equations may be used. P = k. A or P 1 /P 2 = A 1 /A 2
The first equation is in the direct variation form of the relationship. The second equation shows the ratio form of the relationship.
The word inverse means inverted or turned upside down, so if we are told that peaches vary inversely as apples, we remember that the relationships are the upside down form of the direct variation equations. P = k/A or P 1 /P 2 = A 2 /A 1
Some statements of variations give the relationship between three or more variables. The words varies jointly imply a sort of double direct variation that has only one constant of proportionality. Thus, the statement that peaches vary jointly as apples and raisins implies the following relationships: P = k. AR or P 1 /P 2 = A 1 R 1 / A 2 R 2
If we are told that peaches vary inversely as apples and raisins, we must invert the variables on both equations. P = k/AR or P 1 /P 2 = A 2 R 2 / A 1 R 1
In some relationships we have both direct and inverse variations in the same statement. For instance, the statement that girls vary inversely as boys and directly as teachers implies the following equations. G = k. T/B or G 1 / G 2 = B 2 T 1 / B 1 T 2
We note that either the variation form of the equation or the ratio form of the equation may be used. It is helpful to know how to use both forms, because both approaches will be encountered in advanced courses in math and science.
Example: The number of girls varied inversely as the number of boys and directly as the number of teachers. When there were 50 girls, there were 20 teachers and 10 boys. How many boys were there when there were 10 girls and 100 teachers? Work the problem twice; first use the variation form and then use the ratio form.
Answer: B = 250
Example: Strawberries varied jointly as plums and tomatoes. If 500 strawberries went with 4 plums and 25 tomatoes, how many plums would go with 40 strawberries and 2 tomatoes? First work the problem using the variation form, and then work it again using the equal ratio form.
Answer: 4 plums
In lesson 95 we found that the solution to the system 6 x – y = 5 xy = 4 Required the solution to the quadratic equation. When working this problem we can see that it can be factored to get two real solutions. If a quadratic equation cannot be solved by factoring, we can always find the solutions by using the quadratic formula.
Example: Solve x – 2 y = 3 xy = 6
Answer: (3/2 + √ 57/2, -3/4 + √ 57/4) (3/2 - √ 57/2, -3/4 - √ 57/4)
HW: Lesson 96 #1 -30
- Slides: 16