Variation describes a relationship
Two quantities vary when one changes in relation to the other. A variation statement specifies the form of that relationship, not just the observation that both numbers changed. Direct variation means y is a constant multiple of x: becomes y = kx. The constant k is found from known corresponding values. If y = 18 when x = 6, then k = 18/6 = 3, and the model is y = 3x. The same k must apply to every pair covered by that model.
Direct variation gives a straight-line graph through the origin when x is on the horizontal axis. Doubling x doubles y, and the ratio y/x remains constant for nonzero x. A line is linear but does not express direct variation because it has a fixed addition and does not pass through the origin. Recognising this difference prevents treating every increasing relationship as proportional. The quantities can also have units: if y is cost in naira and x is kilograms, k has units naira per kilogram.
Build and use a direct-variation equation
Suppose four identical notebooks cost ₦2,400 with no fixed service charge. Cost C varies directly with number n, so and k = 600. Seven notebooks cost ₦4,200. The phrase “with no fixed charge” matters: if delivery costs an extra ₦500 per order, the full cost is, which is a partial-variation model. Always identify the quantities held constant, such as price per item, before extending the equation to a new situation.
Direct variation may involve a . If the area A of a square varies directly as the square of its side s, write. Doubling the side multiplies area by four, not two. For a square, k = 1 when consistent units are used, but other shape models may have another constant. If and y = 20 when x = 2, then k = 5 and. At x = 3, y = 45. Read the before calculating a scale factor.
Inverse variation keeps a product constant
Inverse variation means, or y = k/x with x ≠ 0. The product xy stays constant. If six equally productive workers finish a fixed job in ten days, the model gives workers × days = 60 worker-days. Fifteen such workers would take four days. This assumes work can be divided efficiently, working hours are unchanged, and extra workers do not reduce one another's efficiency. Real work may depart from the model, so the assumptions belong to the interpretation.
An inverse graph is not a straight line when plotted against x. It is a hyperbola, with forbidden x = 0. If k is positive, positive x gives positive y and increasing x reduces y. A useful linear plot is y against 1/x; its is k. Inverse variation with a has the form ⁿ. For an inverse-square model, doubling x makes y one quarter of its former value. Do not confuse inverse square with taking the square root of an inverse.
Joint and combined variation
Joint variation links more than one factor: becomes z = kxy. If z = 24 when x = 3 and y = 4, k = 2, so z = 2xy. If x doubles while y triples, z becomes six times its original value. Combined direct and inverse variation might be, giving. The wording “directly as x and inversely as the square of y” tells you exactly where each quantity and belongs.
For, suppose z = 12, x = 3 and y = 2. Then, so k = 16. At x = 5 and y = 4, z = 16 × 5/16 = 5. Another route is a ratio. This avoids recalculating k, but it is only safe after the underlying model is clear. Write the direction of every ratio explicitly so that the inverse factor is not accidentally reversed.
Partial variation includes a fixed part
When y is partly constant and partly varies directly with x, write. Two distinct observations determine a and b. A transport charge is ₦1,200 for 4 km and ₦1,800 for 7 km. Subtracting the equations gives 3b = 600, so kilometre. Substituting gives. The model is, and a 10 km trip costs ₦2,400. The fixed part is the vertical intercept; the variable rate is the .
A negative fitted intercept is not automatically impossible algebraically, but it may make the model unsuitable near zero. State the range of inputs for which the relationship is intended. Likewise, measurements that are rounded may make a ratio approximately constant rather than exactly constant. In a theoretical exercise use the supplied exact assumptions; in data interpretation inspect whether variation is an exact rule, an approximation or merely a suggested trend. A table of increasing numbers alone does not establish direct proportionality.
Percentage effects and checking
If and x rises by 10%, multiply x by 1.10, so y is multiplied by 1.10² = 1.21: a 21% increase. If and x rises by 25%, y is multiplied by 1/1.25 = 0.8, a 20% decrease. Equal percentage changes in related quantities are not guaranteed. Self-check: if y varies inversely as x and y = 9 when x = 4, then k = 36 and y = 6 when x = 6. Write the equation first, calculate second, and verify that the direction and size of the change agree with the stated model.
Distinguish a constant from a changing input
The constant of variation belongs to one specified model and set of units. If a cost is measured in naira per kilogram, changing kilograms to grams changes the numerical constant even though the physical relationship is unchanged. With C = 600m for m in kilograms, the equivalent formula for mass g in grams is C = 0.6g. Unit conversion does not create a new physical price.
A table should support the stated relationship. Pairs (2, 6), (4, 12) and (5, 15) share y/x = 3, so they fit y = 3x. Pairs (2, 6), (4, 10) and (5, 12) instead; the increasing trend alone does not establish direct variation. Check a ratio for direct variation, a product for inverse variation, and the stated compound expression for joint variation. If several models are possible from too little information, do not claim the data determine a unique one.