Algebra: Chapter 12, Lesson 7, page 565.
Joint and Combined Variation
An equation of the form `z = kxy`, where `k` is a constant, expresses joint variation. An equation of the form `z = (kx)/y` expresses combined variation. There are 2 steps involved when solving these equations.
- Find `k`, by using all other information given. If there are 2 other variables, then values for those variables must be provided.
- Using the value of `k` that you calculated, and just one of the other variables with NEW values, you can solve for the unknown variable.
Examples 1, 2 and 3 in the textbook walk you through the steps as well.
This link from purplemath.com has more info too.
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Algebra 1a: Chapter 9, Lesson 5, page 417 (day #2).
Inequalities in 2 variables
Given an inequality, treat it as an equality and using the `x` and `y` intercepts, find the solution to the equality. Plot it on your graph paper.
- If the inequality is just a `<`, or `>` problem, then the boundary line (the line you draw connecting the dots) will itself be dotted or dashed. This mean that the points on the line are NOT part of the solution.
- If the inequality has a `≤` or `≥`, then the line will be solid, signifying that the line is part of the solution.
There are 2 ½ planes on the graph, one side of the boundary line that belongs to the solution set (this side will be shaded as part of the solution) and the other side of the line that does not satisfy the inequality.
Now to figure out what ½ plane to shade, pick a point [I like to pick `(0, 0)` or `(1, 1)`] and try those `(x, y)` values in the inequality.
- If the point chosen makes the inequality TRUE, then shade that part of the plane.
- If the point chosen does not satisfy the inequality, then shade the OPPOSITE side ½ plane.
The textbook is actually pretty good in this area, see pages 417-419 for good examples. Purplemath.com has these examples as well.
Two of tonight’s homework problems solved by MrE are here! Just click it