3 Algebra

Learning objectives

33B Change the subject of an equationChange the subject of an equation.33C Substitute values with unitsSubstitute numerical values into algebraic equations using appropriate units for physical quantities.33D Solve simple algebraic equationsSolve simple algebraic equations.

Changing the subject rewrites an equation so the required symbol stands alone while the relationship remains equivalent. Apply inverse operations to both sides, preserving brackets and powers.

Starting relationship	Required subject	Equivalent form
$c=a/v$	$v$	multiply by $v$ , then divide by $c$ : $v=a/c$
$r=d/t$	$t$	$t=d/r$
$B=m/h^2$	$m$	$m=Bh^2$
$B=m/h^2$	$h$	$h=\sqrt{m/B}$ for a positive physical height

Urine concentration is $c=a/v$ . With amount $a=600$ milliosmoles and maximum concentration $c=1400$ milliosmoles dm⁻³, first rearrange to $v=a/c$ , then calculate $v=600/1400=0.429$ dm³, or about 429 cm³.

Moving a term across an equals sign is shorthand for applying an inverse operation to both sides. Do not change a sign or invert a quantity without showing the operation that keeps the equation balanced.

Substitution replaces each symbol with its numerical value. Before calculating, convert quantities to the units required by the equation and use brackets so powers and denominators apply to the intended value.

$BMI = mass in kg / (height in m)^2$

Move	BMI example
identify symbols and required units	mass = 60 kg; height must be in metres
convert units	165 cm = 1.65 m
substitute with brackets	$BMI=60/(1.65)^2$
calculate in operation order	$(1.65)^2=2.7225$ , then $60/2.7225=22.0$
interpret only after calculation	22.0 lies in the stated healthy category

Do not square only part of a substituted height or mix centimetres with a formula defined in metres. Units are part of the input, not decoration added after arithmetic.

Solving an equation finds the numerical value of an unknown that makes both sides equal. Undo operations in reverse order and perform the same operation on both sides.

Equation	Balanced steps	Solution check
$3x+6=24$	subtract 6: $3x=18$ ; divide by 3: $x=6$	$3(6)+6=24$
$y/5-2=4$	add 2: $y/5=6$ ; multiply by 5: $y=30$	$30/5-2=4$
$2z^2=50$	divide by 2: $z^2=25$ ; take the relevant root	for a positive biological length, $z=5$

When an equation contains measured quantities, retain units through the solution and judge whether negative or alternative roots make physical sense in that context.

Changing the subject produces a formula in symbols; solving produces a value after known quantities are supplied. Always substitute the result back into the original equation to detect an arithmetic or sign error.

Next Topic

—4 GraphsRead

3 Algebra

Learning objectives

Rearrange an equation before using it

Substitute values only after aligning units

Solve a simple equation by undoing operations

Next Topic