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This is Variables, Equations and Economic Formulas.

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This walkthrough uses a generic synthetic narrator, not a
recording of Craig Wright.

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Pause whenever you need time to calculate.

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The original lesson, written walkthrough and exercises
accompany this video.

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Letters stand for quantities.

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Algebra does not replace arithmetic with a new kind of mystery.

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It lets us describe the same calculation before choosing
particular numbers.

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If quantity changes, the formula tells us how total cost
changes.

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A letter needs a definition just as a number needs a unit.

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Here q counts units and C measures pounds, so the coefficient
six has the meaning pounds per unit.

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A coefficient is the number multiplying a variable.

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Constants are fixed only within the stated model: a different
supplier or a different period could have a

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different fixed charge or unit cost.

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Expressions, equations and solutions.

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An expression has no equals sign and does not by itself ask you
to find a particular value.

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An equation adds a condition.

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Our example asks which value of x makes three times x plus
seven equal twenty-two.

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The answer is not established merely by producing a number; it
must satisfy the condition when substituted

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back.

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"Linear" will later connect to a straight-line graph.

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For now, notice that x squared and a product such as x times y
would take us beyond the simple linear

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relationships in this lesson.

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We begin with one unknown so each step is easy to inspect.

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Brackets and the order of operations.

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Brackets show which pieces belong together.

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Three times the whole expression x plus two means three copies
of x and three copies of two.

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Missing the second multiplication is a common error.

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Like terms contain the same variable part, so four x plus two x
counts six copies of x.

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A plain two is a different kind of term and stays separate.

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To see why, try x equal to ten: four x plus two gives
forty-two, whereas six x gives sixty.

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For division involving a whole expression, use brackets
explicitly, as in (C minus F) divided by v.

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Substitution turns a formula into arithmetic.

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Write a separate substitution line rather than trying to jump
directly to the answer.

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This makes the chosen value and the order of operations
visible.

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Six times four comes before adding eighteen because
multiplication takes priority over addition.

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The unit check is equally important: six pounds per unit times
four units gives twenty-four pounds, which

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can legitimately be added to the eighteen-pound fixed charge.

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If your formula tried to add pounds directly to a count of
units, that would signal a modelling problem even

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if the arithmetic were possible.

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Substitution is also the main tool for checking an equation
after solving it.

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Preserve the balance of an equation.

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Imagine the equals sign as a balanced relationship.

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Changing just one side generally destroys that balance, while
applying the same valid operation to both

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sides preserves it.

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We undo addition before multiplication here because seven was
added after three multiplied x.

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Subtracting seven removes that outer operation, leaving three
x.

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Dividing by three isolates x.

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"Move it across and change the sign" can be a shorthand, but it
hides the reason the step works.

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Write the operation on both sides while learning.

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Dividing by zero is never valid, and dividing by an unknown
expression requires checking whether it could be

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zero.

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Unknowns can appear on both sides.

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The same balance rule works when x appears on both sides.

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Subtracting two x removes the right-hand variable term and
leaves three x on the left.

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We then proceed as before.

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The checking stage should use the original equation, because an
accidental mistake in an intermediate line

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could otherwise be carried into the check.

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Both original sides evaluate to thirty-two, so the condition
holds.

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You could instead subtract five x first and work with negative
coefficients.

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That route is also valid.

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Choose a clear route, but remember that correctness comes from
preserving equality rather than following one

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compulsory sequence.

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Inequalities describe ranges of answers.

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An equation often identifies a particular value, while an
inequality can identify a whole range.

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The strict symbols exclude the boundary; the symbols with the
extra line include it.

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The sign reversal is easiest to understand on the number line.

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Two is less than five, but multiplying both by minus one gives
minus two, which is greater than minus five.

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Negative multiplication reverses order.

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In our algebra example, division by minus two therefore turns
"less than" into "greater than".

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Test a value from your proposed range in the original
inequality.

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Zero works here, while minus four does not, which supports the
direction we found.

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Worked example: turn a cost question into an equation.

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We already know the structure of this cost model, but now the
total is known and quantity is unknown.

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State q clearly before forming the equation.

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The setup charge appears once, while six pounds is charged for
each unit.

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Subtracting eighteen pounds removes the fixed component from
both sides.

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This leaves seventy-two pounds available to cover the
quantity-dependent component.

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Do not divide ninety by six immediately, because that would
incorrectly treat the setup charge as if it

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purchased units.

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Keep the economic interpretation beside the algebra: six pounds
per unit multiplied by q units must equal

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the remaining seventy-two pounds.

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Worked example: solve, check and interpret.

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Dividing the remaining seventy-two pounds by six pounds per
unit yields twelve units.

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The pounds cancel in the rate calculation, leaving a count of
units rather than another money total.

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Now rebuild the original bill: twelve units cost seventy-two
pounds, and adding the eighteen-pound setup

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charge gives ninety pounds.

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This independent reconstruction is stronger than simply
repeating the division.

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We also check the permitted domain, meaning the values the
variable is allowed to take.

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If the workshop sells only complete units, a negative answer or
a fraction of a unit would need further

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interpretation rather than automatic acceptance.

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Worked example: model revenue, cost and profit.

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The Greek letter pi is simply a label for profit here; it is
not the circle constant in this formula.

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Revenue is money received from sales.

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Variable cost changes with output, while fixed cost is held
constant over the modelled range.

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Subtract the whole cost expression from revenue, which is why
the brackets matter.

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Removing the brackets changes both cost terms to negatives.

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Collecting eighteen q minus eleven q leaves seven q, then
subtract the fixed eighty-four pounds.

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Each additional unit contributes seven pounds towards fixed
cost and then profit, assuming every unit

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produced is sold at the stated price.

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Worked example: meet a minimum profit target.

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The phrase "at least" includes the target itself, so we use
greater than or equal to rather than a strict

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inequality.

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Adding eighty-four to both sides gives seven q greater than or
equal to one hundred and forty.

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We divide by positive seven, so the direction stays unchanged.

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Twenty is the smallest permitted whole quantity.

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Reconstructing revenue and cost at twenty verifies the
boundary.

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At nineteen units, the formula gives forty-nine pounds of
profit, below the target, so nineteen is

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insufficient.

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The conclusion depends on the model assumptions, including
constant prices, constant unit costs and no

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unsold output or capacity restriction.

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Rearrange formulas before inserting numbers.

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The letters now represent general quantities: C is total cost,
F is fixed cost, v is variable cost per unit

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and q is quantity.

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The same balance operations apply even when the numbers are not
yet supplied.

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Brackets around C minus F are essential: the whole difference
is divided by v or q.

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Making a general formula first can reduce repeated work when
several scenarios use the same structure.

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State the non-zero condition for each denominator.

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If no units are produced, total cost alone cannot reveal
variable cost per unit, because the variable-cost

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term contributes nothing regardless of its coefficient.

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Mathematical answers need domain checks.

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Suppose an algebraic lower bound is six point two and products
cannot be divided.

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Six is too small, so round upwards to seven, rather than using
ordinary nearest-integer rounding.

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An upper bound of six point two would allow at most six whole
products.

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Context decides the direction.

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Sometimes simplification removes the unknown entirely.

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If it leaves zero equals three, no value can satisfy the
original equation.

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If it leaves zero equals zero, every value in the permitted
domain satisfies it, provided no earlier

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operation excluded a valid case.

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Checking these outcomes prevents a calculator-looking answer
from replacing actual reasoning.

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An algebra routine you can explain.

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The aim is not to memorise a separate trick for each question.

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A small set of balance rules solves many problems once the
situation is represented accurately.

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Read words such as "exactly", "at most" and "at least"
carefully: they determine whether you need equality

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or an inequality, and which direction it faces.

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Keep one operation per line until the process feels natural.

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For the exercises, retain your working even when the form asks
only for a boundary or a number.

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You should be able to say why the result works and why values
outside an inequality range fail the

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condition.