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This is Coordinates, Straight Lines and Feasible Regions.

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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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A graph connects two quantities.

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A graph is another language for the relationships we have been
calculating.

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One position records two values at once, allowing us to see
patterns that are harder to notice in a list.

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Start every drawing by naming the quantities on the axes.

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"Quantity, in kilograms" and "cost, in pounds" are more useful
labels than x and y alone.

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We conventionally put x horizontally and y vertically, but
economic graphs may use other variable names or

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particular axis conventions.

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Read the labels rather than assuming.

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Every numerical graph in this lesson is an invented teaching
model, not evidence about an actual market.

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Ordered pairs and consistent scales.

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"Ordered" matters because reversing the entries usually places
a point somewhere else.

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Three across and seven up is different from seven across and
three up.

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If negative values are allowed, move left for negative x and
down for negative y.

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Choose scales that fit the relevant values comfortably on the
page.

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For example, each horizontal tick could represent one unit
while each vertical tick represents five pounds.

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That is perfectly valid when labelled clearly.

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However, switching halfway along one axis from increments of
five to increments of ten would distort an

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ordinary linear graph unless a special break or different scale
were explicitly explained.

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Build points from a formula.

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A small value list is a bridge between algebra and a graph.

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Substitute each horizontal value into the formula and preserve
the matching pair.

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Three points are useful because the third can reveal an
arithmetic or plotting error, even though two

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distinct points determine a straight line.

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If the variable can take every intermediate value, connect the
points with a straight line over the

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permitted range.

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If only whole-number quantities are possible, individual
admissible points have the literal meaning; the

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connecting line is a useful model guide.

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Do not extend a line beyond the domain without checking whether
the formula still applies there.

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Slope means vertical change per horizontal change.

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The subscripts simply label the first and second points.

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They do not mean multiplication or powers.

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We calculate the vertical difference and divide it by the
horizontal difference.

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In this example, moving three units right raises the vertical
value by six, so each horizontal unit

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corresponds to two vertical units.

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You may subtract the first point from the second or reverse
both subtractions; either gives the same slope.

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Reversing only one subtraction gives the wrong sign.

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Keep units: if y is cost in pounds and x is quantity in units,
the slope is pounds per unit, not a unit-free

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number.

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Intercepts locate the crossings with axes.

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The two intercepts require different substitutions.

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On the vertical axis the horizontal coordinate is zero, so set
x to zero.

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On the horizontal axis the vertical coordinate is zero, so set
y to zero.

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In our example the crossings are the points zero comma ten and
five comma zero.

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The slope is minus two, meaning a one-unit horizontal increase
is associated with a two-unit vertical

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decrease.

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An algebraic intercept may lie outside the meaningful domain.

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A cost line with a positive fixed charge and positive slope,
for instance, might have a negative horizontal

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intercept that does not describe a feasible production
quantity.

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Positive, negative, zero and undefined slopes.

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Do not confuse a negative slope with a negative value.

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A falling line can remain entirely above the horizontal axis.

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Similarly, a rising line can include negative vertical values.

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Slope describes how values change, not where the line sits.

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A horizontal line has no vertical change, so its slope is zero
divided by a non-zero horizontal change.

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A vertical line has the opposite issue: the horizontal
denominator is zero, so its slope is undefined rather

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than zero.

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Finally, stretching the drawing vertically can make the same
relationship look steeper.

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Numerical slopes and stated units protect against misleading
visual comparisons.

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Movement along a line versus a changed line.

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Consider a cost relationship with a fixed charge and a constant
per-unit charge.

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Ordering more units under the same charges moves us along the
existing cost line.

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Increasing the fixed charge changes the cost associated with
every quantity, so the entire line shifts

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upwards.

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Increasing the per-unit charge instead changes how quickly cost
rises with quantity; the line becomes

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steeper when the axis scales stay fixed.

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This distinction is important because an observed point moving
upwards does not by itself tell us which

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mechanism occurred.

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We need to know whether quantity changed, the underlying terms
changed, or both changed at once.

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Worked example: plot an illustrative cost line.

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Draw quantity in kilograms horizontally and total cost in
pounds vertically.

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A horizontal range from zero to six and a vertical range from
zero to thirty will include all three points.

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Mark equal steps on each axis, then locate the calculated
pairs.

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The zero-quantity point represents the fixed charge built into
this illustrative formula; whether such a

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charge would really be payable with no delivery depends on the
actual service terms.

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Here it is part of the stated model.

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Because kilograms are divisible, the relationship includes
intermediate quantities.

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Join the points with a straight line over the non-negative
quantity range being considered.

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Worked example: interpret the slope and intercept.

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The slope calculation recovers the coefficient of q, which is a
useful check that our plotted values are

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consistent with the formula.

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Four additional kilograms raise cost by twelve pounds, so each
kilogram adds three pounds.

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The intercept of twelve pounds is the model's fixed charge, not
the price per kilogram.

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Solving for a horizontal intercept produces minus four
kilograms.

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This is a valid feature of the unrestricted mathematical line,
but it is not a possible delivery mass.

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Draw the meaningful non-negative part as the model, and explain
rather than physically interpreting any

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extension into negative quantities.

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Worked example: draw a budget boundary.

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We are now graphing two goods rather than a quantity and its
cost.

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Put beans on the horizontal axis and rice on the vertical axis.

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Each combination has a total cost: six pounds times bean
kilograms plus four pounds times rice kilograms.

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The budget allows spending less than or exactly forty-eight
pounds, which explains the less-than-or-equal

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sign.

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To draw the boundary, temporarily replace that sign with
equality.

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Set y to zero: six x equals forty-eight.

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Divide both sides by six: x equals 48 divided by 6 equals 8.

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Set x to zero: four y equals forty-eight.

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Divide both sides by four: y equals 48 divided by 4 equals 12.

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Join the intercepts eight comma zero and zero comma twelve.

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Worked example: choose the affordable side.

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A feasible combination satisfies every stated restriction.

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The first test point spends thirty-two pounds and leaves
sixteen pounds unused, so it lies inside the

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affordable region.

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The second spends fifty-two pounds and exceeds the budget by
four pounds.

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The origin is also affordable because buying neither good costs
zero in this model.

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Use a solid boundary because equality is included: combinations
spending exactly forty-eight pounds are

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allowed.

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Rearranging the boundary gives four y equals forty-eight minus
six x, then y equals twelve minus one point

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five x.

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Its slope means an extra kilogram of beans requires giving up
one point five kilograms of rice when spending

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remains at the budget limit.

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Several constraints mean an intersection.

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The additional restriction might represent a five-kilogram
availability limit in this invented scenario.

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Draw the vertical line x equals five and retain only points to
its left, including the line itself.

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Then keep only points that also satisfy the budget and
non-negativity conditions.

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The word "and" is important: satisfying one constraint is
insufficient.

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The four listed corners outline the intersection.

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Fractional coordinates are acceptable here because kilograms
are divisible.

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With indivisible items, only permitted whole-number points
within the region would describe actual choices.

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A feasible region shows what is allowed; it does not identify a
best choice until an objective or preference

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is specified.

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Read a graph without claiming too much.

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A graph can be arithmetically accurate while encouraging a
mistaken impression.

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Starting the vertical axis near the smallest plotted value can
make a modest difference look dramatic, so

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inspect the tick labels rather than judging height alone.

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Distinguish a theoretical relationship based on stated
assumptions from a scatter of observed values.

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Observed quantities may move together because a third factor
affects both, because influence runs in the

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opposite direction, or because of selection and chance.

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Interpolation uses the relationship within its supported range;
extrapolation extends beyond that range.

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Neither a straight line nor a neat pattern guarantees that such
an extension will remain valid.

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A graph-reading and drawing checklist.

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Connect this routine to the earlier lessons.

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Units tell you what each axis means.

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Algebra supplies the points, intercepts and constraints.

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Ratios underlie slope, while percentage changes offer a
different way to compare movements when the base is

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known.

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For an inequality with a strict symbol, use a dashed boundary
to show that equality is excluded; for an

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inclusive symbol, use a solid boundary.

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A test point should not lie on the boundary, because it must
distinguish the two sides.

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In the exercises, calculate from the stated relationships
rather than judging a sketch by eye, and keep any

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causal conclusion within the evidence actually supplied.