Welcome back. In the previous lesson, you used exponent laws to simplify algebraic expressions, and you practised showing intermediate equal expressions rather than jumping straight to an answer. That habit now becomes the main skill.
This lesson completes the Algebra Fluency for Mathematical Reasoning module. You will learn how to write algebraic working as a derivation: a sequence of statements in which every line follows validly from the previous one. By the end, you should be able to make your method clear enough that an examiner can see both what you did and why it preserved equality.
Equality is a balance, not an instruction
An equals sign says that the expression on its left has the same value as the expression on its right. It does not mean “the answer comes next.”
For example,
states that the left and right sides balance. To remove the , subtract from both sides:
The balance is preserved because the same quantity was removed from each side. Simplifying gives
This is the Subtraction Property of Equality:
If two quantities are equal, subtracting the same quantity from both leaves equal quantities.
The same principle applies when you add, multiply, or divide by the same permitted non-zero quantity.
How to Solve Equations using Properties of Equality
Watch How to Solve Equations using Properties of Equality from Algebra-1 with Mr. Peters. It focuses on an important exam-writing idea: the stated reason must describe the change from one line to the next.
Watch the setup for the purpose of equality properties and a common error in giving reasons. Then watch one derivation, following how the presenter distinguishes distributing, simplifying, subtracting from both sides, adding to both sides, and dividing both sides.
A derivation is therefore more than a list of calculations. It is a compact argument that says:
- Here is the equation or expression I started with.
- Here is one valid change.
- Here is the resulting equivalent statement.
- Here is the reason for that change.
Equivalent expressions and equivalent equations
The word equivalent has a slightly different focus depending on what you are working with.
Expressions
Expressions do not contain an equals sign, such as
Equivalent expressions have the same value for every permitted value of the variable:
This is a simplification derivation.
Equations
Equations do contain an equals sign, such as
Equivalent equations have exactly the same solutions. A valid solving derivation must preserve the equation’s balance at every stage, so the final value solves the original equation too.
This distinction helps avoid a common weak presentation. When simplifying an expression, you may write a vertical chain of equal expressions. When solving an equation, every line must remain an equation:
Do not compress unrelated pieces into a false chain such as
The expression is not generally equal to , so that line does not communicate valid reasoning.
The core reasons you need
You do not always need a formal property name in a short exam question, but you should know what makes each line valid. If a question asks you to “show working,” “justify,” or “give reasons,” these are the labels that make your derivation precise.
| What changed? | Suitable reason | Example |
|---|---|---|
| A bracket was expanded | Distributive property | |
| Like terms or ordinary numbers were evaluated | Simplify / combine like terms | |
| Terms were reordered | Commutative property | |
| Terms were regrouped | Associative property | |
| The same quantity was added to both sides | Addition property of equality | |
| The same quantity was subtracted from both sides | Subtraction property of equality | |
| Both sides were multiplied by the same quantity | Multiplication property of equality | |
| Both sides were divided by the same non-zero quantity | Division property of equality | |
| A known equal quantity replaced another | Substitution property of equality | If , then becomes |
Two cautions matter particularly in exams:
- “Simplify” is not the same as “subtract from both sides.” If you turn into , you have simplified the left side only; you have not performed an operation on both sides.
- Division requires a non-zero divisor. Dividing both sides by is valid; dividing by an expression that might equal needs care.
Basic number properties & how to tell them apart | Purplemath
Read Purplemath’s worked examples to see the same algebra you already know written with a justification beside each line. Its main value here is the sharp distinction between reordering, regrouping, distributing, and simplifying.
In the passage beginning “Once you've learned these properties...,” read the first worked simplification from the prompt “Simplify 3a-5b+7a” through its explanation. Follow the first derivation, noting that moving a term is different from combining it. Then continue to the next two worked prompts, “Simplify 23+5x+7y-x-y-27” and “Simplify 3(x+2)-4x.” Focus on matching each stated property to the exact change made in that line.
A well-written expression derivation
Begin with a simplification problem:
A clear derivation does not try to do everything at once. First remove the bracket, then collect like terms:
Therefore,
There is often more than one correct route. For instance, you could combine before combining the result with . What matters is that every line remains equal to the previous line.
Notice the disciplined use of signs. The term stays negative as it is moved to the end. One reliable way to protect yourself from sign errors is to view subtraction as addition of a negative:
You do not normally need to write that expanded sign form in an exam, but it can help you decide what may be rearranged safely.
A complete equation derivation
Now apply the same clarity to an equation:
The aim is to isolate , while keeping the equation balanced. The derivation below deliberately shows the operation on both sides before simplifying it.
The final conclusion is
Why show the longer “both sides” lines?
You may sometimes see a solution go directly from
to
That step is valid, but it hides the action: subtracting from both sides. In many exams, this shorter form is acceptable once your working is consistently clear. When you are learning, when marks are allocated for reasoning, or when a step feels difficult, write the operation explicitly:
It makes the balance visible and gives you an unambiguous reason to write.
Put the reason in the right place
A reason explains the movement from the line above to the line beside it. It does not describe an operation you plan to do next, and it does not merely name something that happened somewhere nearby.
Consider these lines:
The first change uses the distributive property. The second change is simplification, because became . You did not add to both sides, so “Addition Property of Equality” would be the wrong reason for that second transition.
Use this quick diagnostic for each line of a derivation:
- Did I expand or factor? Use the distributive property.
- Did I only calculate numbers or collect like terms already on one side? Use simplify or combine like terms.
- Did I add, subtract, multiply, or divide both sides by the same quantity? Name that equality property.
- Did I merely alter the order or grouping of terms? Use commutative or associative property if a formal reason is required.
- Is the new line genuinely equal to the previous one?
A compact exam method
When working under time pressure, aim for clarity rather than maximum length.
- Copy the original expression or equation accurately. Brackets and negative signs are part of the mathematics.
- Make one meaningful change per line. Expand brackets before attempting to combine the resulting terms.
- Keep both sides visible when solving an equation. Do not “move” a term without showing its inverse operation.
- Write a reason when requested, or when it makes a major step clear.
- Finish with a conclusion, such as , rather than leaving the result embedded in calculation.
- Check a solution by substitution when time allows.
For the equation above, substitute into both sides:
and
Both sides agree, so the derived solution checks successfully.
A useful final presentation check is to cover the reasons column, look at each pair of consecutive lines, and ask whether you can identify the exact change. If you cannot, the derivation probably skips too much working.
Key takeaways
A logical algebraic derivation is a sequence of equivalent expressions or equations. Every line must preserve value; when solving an equation, every step must preserve the balance and therefore the solution set.
The most important habits are:
- expand, simplify, and solve in separate visible steps;
- apply inverse operations to both sides of an equation;
- match each justification to the transition it explains;
- preserve signs and brackets carefully;
- conclude clearly and check by substitution when possible.
The next module moves from algebraic manipulation to straight lines and simultaneous equations. You will begin by calculating the gradient of a straight line from points or from a graph, using the same disciplined line-by-line communication developed here.
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