Algebra Revision Notes

Understanding Variables and Expressions

In mathematics, a variable is a symbol (usually a letter like $x$, $y$, or $a$) that represents an unknown numerical value. An expression is a combination of numbers, variables, and mathematical operations (like addition, subtraction, multiplication, and division). For example, $2x + 5$ is an expression where $x$ is a variable. A formula is a type of equation that expresses a relationship between different quantities, often used in specific contexts. For example, the area of a rectangle is given by the formula $A = l \times w$, where $A$ is the area, $l$ is the length, and $w$ is the width.

The Process of Substitution

Substitution means replacing a variable in an expression or formula with a given numerical value. The steps are:

1. Identify the variables in the expression or formula.
2. Note the numerical values assigned to each variable.
3. Carefully replace each variable with its corresponding numerical value. It is good practice to use brackets around substituted negative numbers to maintain the correct order of operations and prevent errors.
4. Perform the calculations using the correct order of operations (BODMAS/PEMDAS: Brackets, Orders (powers/roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right)).

Handling Signs and Operations

Be especially careful when substituting negative numbers. For example, if $x = -4$ and you need to evaluate $3x$, it becomes $3 \times (-4) = -12$. If the expression is $x^2$, it becomes $(-4)^2 = (-4) \times (-4) = 16$. A common mistake is writing $-4^2 = -16$, which implies $-(4^2)$. Always assume an implied multiplication when a number is next to a variable (e.g., $5a$ means $5 \times a$) or when variables are next to each other (e.g., $xy$ means $x \times y$). Showing your substitution step before calculation helps prevent errors.

Understanding Variables and Substitution

In mathematics, variables are letters (like x, y, a, b) that represent unknown or changing numerical values. Substitution is the process of replacing these variables with specific numerical values. For example, if a formula describes the cost of something based on a quantity, you substitute the actual quantity to find the cost. This skill is vital for solving real-world problems and algebraic equations.

Order of Operations (BIDMAS/PEMDAS)

When substituting values, always strictly follow the order of operations:

• Brackets (or Parentheses)
• Indices (or Exponents)
• Division and Multiplication (from left to right in the expression)
• Addition and Subtraction (from left to right in the expression)

Ignoring this order will almost certainly lead to incorrect answers. For instance, in 2 + 3 × 4, multiplication must be done before addition: 2 + 12 = 14, not 5 × 4 = 20.

Handling Negative Numbers and Fractions

When substituting negative numbers, it's good practice to place them in brackets to prevent errors, especially when squaring or multiplying. For example, if x = -2 and you need x^2, write (-2)^2 = 4, not -2^2 = -4. When substituting fractions, remember your fraction arithmetic rules for addition, subtraction, multiplication, and division.

Steps for Successful Substitution

1. Read the formula or expression carefully.
2. Identify the values given for each variable.
3. Replace each variable with its corresponding numerical value. Use brackets for clarity, especially with negative numbers or when a number is being multiplied by a variable (e.g., 3x becomes 3(5) if x=5).
4. Perform the calculations following the correct order of operations (BIDMAS/PEMDAS). Double-check each step to minimise errors.

Core method

Changing the subject means rearranging a formula so a chosen variable is isolated. Apply inverse operations to both sides and work in the reverse order from the operations acting on the new subject. Keep equations balanced and use brackets when an entire expression is divided. The essential approach is to identify the mathematical structure before calculating, retain exact values where possible, and show enough working for each conclusion to be checked independently.

Worked reasoning and checks

From y = 3x + 5, subtract 5 and divide by 3 to get x = (y − 5)/3. From A = (a + b)h/2, multiply by 2, divide by h, then subtract a to obtain b = 2A/h − a. A final substitution can check the rearrangement. Check the final result against the original information, including signs, units, scale and whether the requested form or degree of accuracy has been supplied.

Basic Principles of Rearrangement

Changing the subject of a formula is like solving an equation, but instead of finding a numerical value, you are isolating a specific variable. The core principle is that whatever operation you apply to one side of the equals sign, you must apply to the other side to keep the formula balanced. Think of the equals sign as a balance point. For example, if you have x + a = b and you want to make x the subject, you subtract 'a' from both sides: x + a - a = b - a, which simplifies to x = b - a. Similarly, if you have xa = b, you divide both sides by 'a': xa/a = b/a, resulting in x = b/a. Always aim to get the target variable by itself, with a positive coefficient of 1.

Order of Operations for Isolation

When rearranging a formula, it's often helpful to think of reversing the order of operations. If you're trying to isolate a variable, first deal with any terms that are added or subtracted to the term containing your target variable. For example, in the formula A = B + C, to make B the subject, you would first subtract C from both sides: A - C = B. After dealing with addition and subtraction, address multiplication and division. If you have Y = 3X, and you want to make X the subject, you would divide both sides by 3: Y/3 = X. Following this systematic approach helps avoid errors when multiple operations are involved. It's like unwrapping a present; you peel back the outermost layers first.

Handling Negative Coefficients and Divisions

Sometimes, after performing operations, your target variable might have a negative sign or be in the denominator. For example, if you get -x = y, you can multiply or divide both sides by -1 to get x = -y. If your target variable ends up in the denominator, such as P = Q/R and you want to make R the subject, you must first multiply both sides by R to get it out of the denominator: PR = Q. Then, you can divide both sides by P to isolate R: R = Q/P. Always ensure the subject variable has a positive coefficient of 1 and is in the numerator at the end of the rearrangement.