lilernoEngineeringOpening Lilerno
Chapter 02 · Guided self-study
Intro + external course studyFind an unknown without changing the meaning of an equation.
Build the idea
An original introduction, worked example and practice, followed by a curated reading sequence. The external courses supply deeper teaching and problem sets.
This guide is an orientation, not a complete university module. Practical work needs suitable facilities, safety review and supervision.
Start with these prerequisites
Shared science foundations
An equation is a statement that two quantities are equal. Engineering algebra lets you find an unknown while keeping that statement true. Start with ordinary numbers before manipulating letters.
Module lens
Rearrange a physical relationship symbolically before inserting numbers, then check the result by substitution.
Choose a positive direction. If right is positive, a move of +5 m followed by −2 m gives a net displacement of +3 m. Distance travelled is 7 m, not 3 m: a negative sign tells you direction, not a negative amount of travel.
For 3x + 4 = 19, subtract 4 from both sides to get 3x = 15. Divide both sides by 3 to get x = 5. Substitute it into the original equation: 3 × 5 + 4 = 19. Never divide by zero, and never change just one side.
Average speed v = d/t relates distance d and positive elapsed time t. Multiplying both sides by t gives vt = d. For nonzero v, dividing by v gives t = d/v. Work with symbols first, then insert compatible units. A ratio such as d/v is a division, not a new quantity you can guess from its letters.
Scientific notation separates a number from a power of ten: 0.003 m = 3 × 10⁻³ m. The exponent −3 means divide by 1,000. When multiplying powers of ten, add exponents; when dividing, subtract them. Conversion should change the label, not the physical size.
See the relationship
Change one quantity, watch the graph respond, then explain the result in your own words.
Predict → change → explain
Change one quantity. Does the equation still balance?
Axes: Time (s) → Distance (m). Bounds may rescale when inputs change.
Time = d/v = 6 s. Check: 0.3 m/s × 6 s = 1.8 m.
Constant average speed for an illustrative journey. This model relates d, v and t; it does not describe acceleration or a real vehicle.
Original Lilerno illustration. Inputs are illustrative; this is not experimental evidence or a design rating.
| Series | Time (s) | Distance (m) |
|---|---|---|
| Constant-speed journey | 0 | 0 |
| Constant-speed journey | 0.075 | 0.0225 |
| Constant-speed journey | 0.15 | 0.045 |
| Constant-speed journey | 0.225 | 0.0675 |
| Constant-speed journey | 0.3 | 0.09 |
| Constant-speed journey | 0.375 | 0.1125 |
| Constant-speed journey | 0.45 | 0.135 |
| Constant-speed journey | 0.525 | 0.1575 |
| Constant-speed journey | 0.6 | 0.18 |
| Constant-speed journey | 0.675 | 0.2025 |
| Constant-speed journey | 0.75 | 0.225 |
| Constant-speed journey | 0.825 | 0.2475 |
| Constant-speed journey | 0.9 | 0.27 |
| Constant-speed journey | 0.975 | 0.2925 |
| Constant-speed journey | 1.05 | 0.315 |
| Constant-speed journey | 1.125 | 0.3375 |
| Constant-speed journey | 1.2 | 0.36 |
| Constant-speed journey | 1.275 | 0.3825 |
| Constant-speed journey | 1.35 | 0.405 |
| Constant-speed journey | 1.425 | 0.4275 |
| Constant-speed journey | 1.5 | 0.45 |
| Constant-speed journey | 1.575 | 0.4725 |
| Constant-speed journey | 1.65 | 0.495 |
| Constant-speed journey | 1.725 | 0.5175 |
| Constant-speed journey | 1.8 | 0.54 |
| Constant-speed journey | 1.875 | 0.5625 |
| Constant-speed journey | 1.95 | 0.585 |
| Constant-speed journey | 2.025 | 0.6075 |
| Constant-speed journey | 2.1 | 0.63 |
| Constant-speed journey | 2.175 | 0.6525 |
| Constant-speed journey | 2.25 | 0.675 |
| Constant-speed journey | 2.325 | 0.6975 |
| Constant-speed journey | 2.4 | 0.72 |
| Constant-speed journey | 2.475 | 0.7425 |
| Constant-speed journey | 2.55 | 0.765 |
| Constant-speed journey | 2.625 | 0.7875 |
| Constant-speed journey | 2.7 | 0.81 |
| Constant-speed journey | 2.775 | 0.8325 |
| Constant-speed journey | 2.85 | 0.855 |
| Constant-speed journey | 2.925 | 0.8775 |
| Constant-speed journey | 3 | 0.9 |
| Constant-speed journey | 3.075 | 0.9225 |
| Constant-speed journey | 3.15 | 0.945 |
| Constant-speed journey | 3.225 | 0.9675 |
| Constant-speed journey | 3.3 | 0.99 |
| Constant-speed journey | 3.375 | 1.012 |
| Constant-speed journey | 3.45 | 1.035 |
| Constant-speed journey | 3.525 | 1.057 |
| Constant-speed journey | 3.6 | 1.08 |
| Constant-speed journey | 3.675 | 1.102 |
| Constant-speed journey | 3.75 | 1.125 |
| Constant-speed journey | 3.825 | 1.147 |
| Constant-speed journey | 3.9 | 1.17 |
| Constant-speed journey | 3.975 | 1.192 |
| Constant-speed journey | 4.05 | 1.215 |
| Constant-speed journey | 4.125 | 1.238 |
| Constant-speed journey | 4.2 | 1.26 |
| Constant-speed journey | 4.275 | 1.282 |
| Constant-speed journey | 4.35 | 1.305 |
| Constant-speed journey | 4.425 | 1.327 |
| Constant-speed journey | 4.5 | 1.35 |
| Constant-speed journey | 4.575 | 1.373 |
| Constant-speed journey | 4.65 | 1.395 |
| Constant-speed journey | 4.725 | 1.417 |
| Constant-speed journey | 4.8 | 1.44 |
| Constant-speed journey | 4.875 | 1.462 |
| Constant-speed journey | 4.95 | 1.485 |
| Constant-speed journey | 5.025 | 1.508 |
| Constant-speed journey | 5.1 | 1.53 |
| Constant-speed journey | 5.175 | 1.552 |
| Constant-speed journey | 5.25 | 1.575 |
| Constant-speed journey | 5.325 | 1.597 |
| Constant-speed journey | 5.4 | 1.62 |
| Constant-speed journey | 5.475 | 1.642 |
| Constant-speed journey | 5.55 | 1.665 |
| Constant-speed journey | 5.625 | 1.688 |
| Constant-speed journey | 5.7 | 1.71 |
| Constant-speed journey | 5.775 | 1.733 |
| Constant-speed journey | 5.85 | 1.755 |
| Constant-speed journey | 5.925 | 1.777 |
| Constant-speed journey | 6 | 1.8 |
Save your place when you finish reading.
Original Lilerno example
A carriage covers 1.8 m at an average speed of 0.3 m/s. Find the elapsed time.
State v = d/t with positive elapsed time.
Multiply by t, then divide by v: t = d/v.
Insert values: 1.8 ÷ 0.3 = 6.
Check by substitution: 0.3 m/s × 6 s = 1.8 m.
Test the model
Open learning, traceable sources
Work in this order. These links open the publisher’s material; free access does not always permit republication.
OpenStax
Start with: Section 2.2: division and multiplication properties of equality
Work through the section examples, then solve three practice equations without looking at the answer. Check each by substitution.
OpenStax
Start with: Signed numbers, fractions and powers
Review any arithmetic operation that slowed you down before continuing to trigonometry.
Recall, then record
Close the explanation and answer these in your own words. Return tomorrow, then again later in the week.
Why must an operation be applied to both sides?
When is division by a variable invalid?
How do displacement and distance differ?
Write and solve three versions of d = vt: unknown d, unknown v and unknown t. Include units and a substitution check for each.
Self-reported tasks, not an assessment of mastery or university credit. Reading a page does not complete a chapter.
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Open learning, traceable sources
Original Lilerno lessons and diagrams, supported by these references. Free access does not always permit republication. Links open the publisher’s material.
Practise inverse operations, then solve equations containing fractions and variables on both sides.
External reading only. Check current book and figure terms before copying; no OpenStax content is redistributed here.
Open source ↗Rights / publisher record ↗Arithmetic, fractions, powers, and the bridge to algebra.
Reference only. Verify the current book and asset licence before reuse.
Open source ↗Rights / publisher record ↗