Rearranging and Solving

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infoWhy this? Solving equations and rearranging formulae develop fluency with equivalence and inverse operations. Fractions and brackets make these processes more general, while substitution checks promote accuracy and mathematical justification.

scheduleWhy now? This unit builds on Year 7 one- and two-step equations and on recent work with expansion and factorisation. It extends those methods to fractional equations, brackets, and simple changes of subject before equations with unknowns on both sides.

An equation is a statement that two expressions are equal, and a solution is any value of the variable that makes the equation true.
The equals sign means “has the same value as”, so whatever you do to one side of an equation you must do to the other side to keep it balanced.
Inverse operations undo each other, for example addition and subtraction are inverses, and multiplication and division are inverses.
A one-step equation can be solved by doing one inverse operation, for example if `x+7=19` then `x=19-7`.
If `ax=b` then `x=b\div a`, so dividing both sides by the coefficient isolates the variable.
A two-step linear equation in the form `ax+b=c` is solved by subtracting `b` first, then dividing by `a`.
Solving can give non-integer solutions; for example if `2x+1=8` then `2x=7` and `x=3.5`.
A fraction is a division, so `\frac{a}{b}` means `a\div b` and can appear as a number or within an algebraic expression.
To solve an equation containing fractions, you can multiply every term by a common denominator to remove the fractions without changing the solution.
When multiplying both sides by a common denominator, every term on both sides must be multiplied (not just the fraction term).
Brackets show that everything inside is treated as a single group, so operations outside the brackets apply to the whole bracketed expression.
The distributive law allows you to expand brackets: `a(b+c)=ab+ac` and `a(b-c)=ab-ac`.
When a negative sign is in front of brackets, it multiplies every term inside: `-(b+c)=-b-c`.
After expanding brackets, like terms can be collected (for example `3x+2x=5x`) to simplify an equation before solving.
A formula is an equation used to show a relationship between quantities, and “changing the subject” means making a different variable the one that is isolated.
Changing the subject in one step means using a single inverse operation to isolate the required variable, for example from `A=lw` you can make `l=A\div w`.
A correct solution can be checked by substituting the value back into the original equation to see if both sides are equal.

rocket_launchYou must be able to

Form a one-step equation from a worded statement by defining the variable and translating the operation (for example “5 more than a number is 12” into `x+5=12`).
Solve one-step equations by applying the inverse operation to both sides and writing the solution clearly (for example `x-4=9` then add 4 to both sides).
Solve two-step equations of the form `ax+b=c` by undoing addition/subtraction first, then undoing multiplication/division; show each step as a balanced equation.
Solve equations involving fractions by multiplying every term on both sides by the lowest common denominator, then solving the resulting integer-coefficient equation.
Solve equations involving brackets by expanding using the distributive law, simplifying by collecting like terms, then isolating the variable.
Handle negative brackets correctly by distributing the negative sign to every term before simplifying.
Change the subject of a formula in one step by identifying the operation linking the subject to the rest of the formula and applying the inverse operation to both sides.
Check a solution by substituting it into the original equation and confirming both sides evaluate to the same number.

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