When x is 8, y is −40. Direct variation means the ratio y/x stays constant, so first find the constant of variation k using the known pair: k = y/x = 10 divided by −2 = −5. Then apply it to the new input: y = kx = (−5)(8) = −40. The model is y = −5x throughout.
Direct variation as a fixed ratio
Direct variation describes a relationship in which two quantities move together at a constant rate. The standard form is y = kx, where k is the constant of variation. The phrase "varies directly" is a promise about proportions: every time x changes by a factor, y changes by the same factor. Double x and y doubles; halve x and y halves, even when the numbers are negative.
The mechanism is division. Because y/x equals the same k for every point on the line, any single known pair calibrates the whole relationship. The known pair here is (−2, 10), which gives k = −5. That single number is a compression of infinitely many predictions. Ask for y at any x, and multiplication supplies it. The graph of this relationship is a straight line through the origin with slope −5, and any point you compute will sit on that line.
The constant itself tells you two things at once. Its magnitude, 5, sets the steepness: each unit of x shifts y by five units. Its sign, negative, sets the direction: when x rises, y falls. Professionals read the sign first because it determines the qualitative story before any arithmetic happens.
Here is a second pair to check the model. When x = 0, y = 0, because any number times zero is zero. This zero point is the signature of direct variation, and its absence is the fastest way to recognize that a relationship is not proportional after all.
Why the sign is the whole story
The negative constant is the detail that separates careful work from careless work. With k = −5, positive inputs produce negative outputs and negative inputs produce positive outputs. The pair (−2, 10) lives in the second quadrant; the new pair (8, −40) lives in the fourth. Students who ignore the sign report 40 and are wrong by 80 units of movement on the number line.
The signs also give a directional check that costs nothing. From x = −2 to x = 8, the input increases by 10 units. A negative constant means the output must decrease by 5 times that amount, or 50 units. Starting from 10 and subtracting 50 lands exactly on −40. This path confirms the answer without recomputing the constant, and it catches sign errors that a single multiplication can hide.
Consider what would change if the constant were positive. With k = 5, the same known pair would be impossible, because (−2, 10) would not satisfy y = 5x. The negative sign is not decoration; it is forced by the data. Every direct variation problem carries this kind of internal consistency requirement, and checking it is part of the solution rather than an afterthought.
The ratio table for x = 8
The table below runs the model across the region relevant to the question, including the known pair and the requested input. Every row multiplies x by −5, so internal consistency is visible at a glance. Values of x between −2 and 8 are shown so you can watch the output cross zero exactly when the input does.
| x | y = −5x | Note |
|---|---|---|
| −4 | 20 | Two units left of the known pair; y rises by 10 |
| −2 | 10 | Known pair from the problem |
| 0 | 0 | Origin crossing; confirms true direct variation |
| 2 | −10 | Output now negative |
| 4 | −20 | Same magnitude as known pair, opposite sign |
| 8 | −40 | Requested value |
| 10 | −50 | One step beyond; y falls another 10 |
The pattern to internalize is the paired symmetry at ±2 and ±4: inputs of equal size but opposite sign produce outputs of equal size but opposite sign. That mirror behavior is a property of any directly proportional relationship and a quick diagnostic when checking work.
How the answer changes when the model is not proportional
Direct variation is the tightest of the proportional models, and its alternatives produce different answers from the same starting pair. The comparison below shows where the y-value at x = 8 lands under each structure. It matters because textbook problems sometimes say "varies directly" when the deeper mathematical structure would be different, and professionals are expected to notice the difference.
| Model | Form | k or parameter | y at x = 8 |
|---|---|---|---|
| Direct variation | y = kx | k = −5 | −40 |
| Linear, nonzero intercept | y = kx + b | k = −5, b = 0 only if proportional | −40 if b = 0; otherwise differs |
| Inverse variation | y = k/x | k = −20 | −2.5 |
| Direct with square | y = kx² | k = 2.5 | 160 |
| Direct with cube | y = kx³ | k = −1.25 | −640 |
The row that catches people is inverse variation. It also uses "when x is −2, y is 10" to pin a constant, but the constant is a product, k = xy = −20, not a ratio. Under inverse variation the answer at x = 8 is −2.5, not −40. The words "varies directly" versus "varies inversely" are the only thing separating these outcomes, so reading them carefully is a real skill, not pedantry.
Four incorrect answers worth knowing
Most wrong answers to this problem are predictable, and each one teaches something about the underlying mechanism. Treat them as red flags to catch in your own work.
Reporting 40 instead of −40: the sign of k was dropped, so the direction of change was ignored. Always check whether the output should rise or fall. Computing 8 divided by negative 2 times 10: this multiplies by the input ratio instead of applying the constant. Direct variation needs k, not a scale factor on the original pair. Setting up 10/2 = y/8 and solving to get 40: the sign was lost when the negative pair was converted to positives, a common transcription error. Assuming y stays 10 because no other information was given: this abandons the model entirely. The constant of variation is the "other information" the problem provides.
A fifth trap appears when a context adds noise. If a table of data is said to be directly proportional but the rows give ratios of 3, 3.2, and 2.8, the table does not describe direct variation, and no single constant will fit every row. The right move is to say so rather than average the ratios and invent certainty. For a clean pair like the one in this problem, the model is exact, and the confidence is justified.
When direct variation appears in real measurement
Direct variation models distance at constant speed, cost at a fixed unit price, and mass from volume at constant density. Each case carries a limit. A car traveling at 60 miles per hour covers 60 miles in one hour and 120 miles in two, exactly proportional only while speed remains constant. A gasoline purchase at a fixed price per gallon scales linearly until taxes, discounts, or minimum purchase charges introduce a fixed fee. The model breaks at the first nonzero intercept.
Temperature conversion is a classic illustration of what is not direct variation. Converting degrees Celsius to Fahrenheit multiplies by 9/5 and adds 32, so the relationship is linear but not proportional. The ratio of the readings is not constant across temperatures, and predictions built on a single pair would fail. Engineers use proportional models first because they are easy to invert, but they verify the zero point before trusting them.
Reverse the direction and the same ratio does the reverse job. If the output is known and the input is the question, divide by k. Direct variation is invertible, so a delivery cost of −40 in this model corresponds to an input of 8. Nothing in the problem's presentation indicates which direction will be requested, which is why the constant deserves to be recorded explicitly before any computation begins.
One historic threshold is worth remembering here for a different reason. The IEEE 754 standard for floating-point arithmetic governs how computers store decimals like −2 and 10, and it has been stable since 1985. That stability is why a script writing y = -5 * x returns the exact answer for these inputs, while the same arithmetic on quantities like 0.1 introduces stored rounding error. The model is exact; the machine representation may not be.
Building a reliable check
The technique generalizes to any pair of points on a directly proportional relationship, and a four-step routine prevents nearly every error.
Read the relationship words first, confirming "varies directly" rather than inversely. Extract the known pair and compute k as y divided by x, keeping the sign of both numbers. Write the model y = kx explicitly with the numeric k before using it. Substitute the requested input, multiply, and state the result with its sign. Run one directional check: confirm the output moves in the direction the constant's sign requires.
For larger problems with three or more pairs, compute k from each pair and compare. A mismatch of any pair flags either a transcription error or a model that is not direct variation. This check costs seconds and eliminates the most common failure mode: silently copying a number with the wrong sign.
Limits of the single-pair answer
The pair (−2, 10) is a single observation, and direct variation is an assumption, not a proof. Within the stated model the answer is exactly −40, with no uncertainty in the arithmetic. Outside the model, the data support no conclusion. If the real relationship had a nonzero intercept, or curved, or changed regime beyond some input, the single pair could not detect it.
That distinction between exactness inside a model and uncertainty outside it is the professional caveat worth carrying forward. Direct variation is a powerful but narrow claim: constant ratio, zero intercept, no thresholds. State it, apply it, and note where it stops holding. For this problem the requested arithmetic is unambiguous, and the verified result is −40.