What is Direct Variation and How to Solve It?
Direct variation is a type of linear relationship between two variables, where one variable changes by a constant factor as the other variable changes. For example, if you drive at a constant speed, then the distance you travel varies directly with the time you spend driving. In other words, the more time you drive, the more distance you cover.
Lesson 5 Homework Practice Direct Variation Answer Key
Direct variation can be expressed using the symbol â, which means \"varies directly with\". For example, if x and y are directly proportional, we can write y â x. This means that as x increases or decreases, y increases or decreases by the same factor.
The constant factor that relates two variables in direct variation is called the constant of variation or the constant of proportionality. It is usually denoted by k. We can find the value of k by using the formula y = kx, where y and x are any pair of values that satisfy the direct variation. For example, if y = 12 when x = 3, then we can find k by plugging in these values into the formula: 12 = k * 3, which gives k = 4. This means that y varies directly with x with a constant of variation of 4.
How to Solve Direct Variation Problems?
To solve direct variation problems, we need to follow these steps:
Identify the two variables that are in direct variation and write them using the symbol â. For example, if the distance d varies directly with the time t, write d â t.
Write the formula for direct variation using the constant of variation k: y = kx.
Use the given information to find the value of k. For example, if d = 100 miles when t = 2 hours, plug in these values into the formula and solve for k: 100 = k * 2, which gives k = 50.
Use the value of k to answer the question. For example, if we want to find d when t = 3 hours, plug in these values into the formula and solve for d: d = 50 * 3, which gives d = 150 miles.
Examples of Direct Variation Problems
Here are some examples of direct variation problems and how to solve them:
Example 1: The amount of water in a tank varies directly with the time it is filled. If it takes 5 minutes to fill 20 gallons of water, how long will it take to fill 60 gallons of water?
Solution: Let w be the amount of water in gallons and t be the time in minutes. Then we have w â t and w = kt. To find k, we use the given information: 20 = k * 5, which gives k = 4. To find t when w = 60, we plug in these values into the formula and solve for t: 60 = 4 * t, which gives t = 15 minutes.
Example 2: The cost of printing a book varies directly with the number of pages. If it costs $12 to print a book with 200 pages, how much will it cost to print a book with 300 pages?
Solution: Let c be the cost in dollars and p be the number of pages. Then we have c â p and c = kp. To find k, we use the given information: 12 = k * 200, which gives k = 0.06. To find c when p = 300, we plug in these values into the formula and solve for c: c = 0.06 * 300, which gives c = $18.
Lesson 5 Homework Practice Direct Variation Answer Key
If you are looking for an answer key for lesson 5 homework practice on direct variation, you can download it from this link. This answer key contains solutions for various direct c481cea774
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