Rearranging and Solving

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infoWhy this? Solving equations develops a rigorous understanding of equality, inverse operations, and equivalent transformations. Function machines provide an accessible model for working forwards and backwards through relationships before these ideas are expressed formally.

scheduleWhy now? This unit follows early algebra and sequences, where substitution and symbolic rules have already been introduced. It now formalises inverse reasoning so that we can solve linear equations and later rearrange formulae and analyse graphs.

A function machine maps an input to an output by applying one or more operations in sequence.
In a one-step function machine, the output is obtained by a single operation such as addition, subtraction, multiplication, or division.
In a two-step function machine, the output is obtained by performing two operations in order, for example first multiplying by a number and then adding a number.
Inverse operations are pairs that undo each other: addition and subtraction are inverses, and multiplication and division are inverses.
To reverse a two-step function machine, you apply the inverse operations in the reverse order to recover the original input from the output.
An equation is a statement that two expressions are equal, and solving an equation finds the value of the variable that makes the statement true.
Adding or subtracting the same quantity from both sides, or multiplying or dividing both sides by the same non-zero quantity, produces an equivalent equation.
A one-step equation involves one operation on the variable, such as `x + a = b`, `x - a = b`, `ax = b`, or `x/a = b`.
A two-step linear equation has the form `ax + b = c`, where `a`, `b`, and `c` are numbers and `a \ne 0`.

rocket_launchYou must be able to

Evaluate the output of a one-step or two-step function machine for a given input, including negative and fractional inputs, showing the intermediate step(s) .
Determine the input that produced a given output by applying inverse operations in reverse order, writing the corresponding working line by line .
Solve a one-step equation by applying the inverse operation to both sides, e.g. from `x + a = b` write `x = b - a`, or from `ax = b` write `x = b/a` .
Solve `ax + b = c` by subtracting `b` from both sides and then dividing both sides by `a`, where `a \ne 0` .

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