Variations

1Summary 重點摘要

Learn direct, inverse, joint variation, solve real proportion problems.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Decide the type of variationVariation describes a fixed relationship between two or more quantities. Catch the key words when reading: 'y varies directly as x' means direct variation, 'y varies inversely as x' means inverse variation; when two variables are involved at once, for example y varies directly as x² and inversely as z, it is joint variation. Choosing the wrong type ruins the whole question, so translate the words into a proportion statement first, confirm how one quantity follows the other, and only then move on to the equation.
2. Step 2: Write the equation with kOnce the type is decided, write the general equation at once: y = kx for direct variation; y = k/x for inverse variation, that is, y varies directly as 1/x; for joint variation, multiply and divide the factors accordingly, for example y = kx²/z. The constant k is called the variation constant, and each question has its own one. Common errors are writing y = -kx for inverse variation, or dropping a square factor such as x²; check the equation word by word against the statement in the question.
3. Step 3: Find k from the given valuesThe question usually provides one pair of matching values, for example 'when x = 2, y = 10'. Substitute this pair into the equation from Step 2, which produces a simple equation in k alone, and solve it. The meaning of this step is that k turns the general relationship into the specific one for this question; k differs from question to question and must never be carried over from a previous problem. If k comes out as a fraction, keep the fraction form, which is more accurate than a decimal and easier to check later.
4. Step 4: State the complete relationshipAfter finding k, substitute it back to write the complete relationship for this question, such as y = 5x or y = 120/x. Many students stop once k is found, but writing the explicit relationship prevents using the wrong formula in the next step. To check it, substitute the original pair of values back into the relationship and see whether both sides are equal; if they are, the relationship is correct, and if not, either k or the equation is wrong and should be re-examined at once.
5. Step 5: Substitute to find the new valueWith the complete relationship written, substitute the new value that the question asks for. In direct variation, when x grows, y grows in the same proportion; in inverse variation, when x grows, y shrinks. Estimating the direction of the answer before substituting therefore reveals a misplaced value at once. Write out the expression after substitution and evaluate it in order instead of jumping steps; in joint variation, align every variable with its position and power, since dropping a square or a divisor is a common loss of marks.
6. Step 6: Check the answerFinally verify on two levels: first, put the new answer together with the known values back into the relationship and confirm the equation holds; second, check that the direction of change matches the type of variation, since a direct variation should never show one quantity rising while the other falls. Also consider the practical meaning of the answer, for example a length or a weight cannot be negative. The check takes only a few seconds but intercepts most computational errors, making it a reliable way to secure marks.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Direct: y = kx, proportional
• Inverse: y = k/x, inversely proportional
• First use known numbers find k
• After find k can find unknown
• Direct x times y also times, inverse x times y half