Algebraic Vocabulary and Substitution

Algebraic Definitions and Vocabulary

To succeed in the TMUA, you must use the correct mathematical terminology. These terms describe the structures used to build and manipulate algebraic models.

1. Formula: A formula represents a relationship between specific physical or mathematical variables. For example, if a rectangle has length $l$, width $w$, and perimeter $P$, then $P = 2l + 2w$ is the formula for the perimeter. It links $P$ to $l$ and $w$.

2. Expression: An expression is a collection of terms without an equals sign. In our rectangle example, $2l + 2w$ is the expression for the perimeter.

3. Terms: These are the individual parts of an expression, separated by addition or subtraction. The expression $2l + 2w$ has two terms: $2l$ and $2w$. Note that an expression can consist of a single term.

4. Factor: A factor of a quantity or expression is a part that divides exactly into it without leaving a remainder. For instance, $xy$ is a factor of $3x^2y^3$, and $(x + 1)$ is a factor of $x(x + 1)(x + 3)$.

Equations, Identities, and Inequalities

It is vital to distinguish between statements that are sometimes true and statements that are always true.

Equations contain an equals sign and are true only for specific values of the variable. For example, $3x + 1 = 7$ is an equation that is true only when $x = 2$.

Identities are true for every possible value of the variable. We often use the identity symbol $\equiv$ instead of a standard equals sign to emphasize this. For example, $3x + 2x - 4x \equiv x$ is an identity because the left and right sides are mathematically identical for any $x$.

Inequalities describe the relative size of two expressions using symbols such as less than ($<$), less than or equal to ($\leq$), greater than ($>$), greater than or equal to ($\geq$), or not equal to ($\neq$). Examples include $x^2 > 4$ or $4x - 3 \geq -2$.

Factors of Algebraic Expressions

When identifying factors of an algebraic term, you must consider all possible combinations of the variables and constants present.

Worked Example: Factors of $p^3q^2$

The factors are: $1, p, q, p^2, pq, q^2, p^3, p^2q, pq^2, p^3q, p^2q^2, p^3q^2$.

Worked Example: Factors of $x(x + 3)(x - 5)$

The factors are: $1, x, (x + 3), (x - 5), x(x + 3), x(x - 5), (x + 3)(x - 5), x(x + 3)(x - 5)$.

Substitution into Algebraic Expressions

Numerical substitution requires strict adherence to the order of operations, known as BIDMAS (Brackets, Indices, Division, Multiplication, Addition, Subtraction).

Worked Example: Simple Substitution

What is the value of $3x^3 - 5(x - y)$ when $x = 2$ and $y = 5$?

1. Substitute the values: $3(2^3) - 5(2 - 5)$.
2. Handle the bracket: $3(2^3) - 5(-3)$.
3. Handle the index: $3(8) - 5(-3)$.
4. Multiply: $24 - (-15)$.
5. Add: $24 + 15 = 39$.

Worked Example: Substitution with Negatives

What is the value of $6x^2 - 3(y^3 - 2x)$ when $x = 4$ and $y = -2$?

1. Substitute the values: $6(4^2) - 3((-2)^3 - 2(4))$.
2. Handle the bracket (index first): $6(16) - 3(-8 - 8)$.
3. Finish the bracket: $6(16) - 3(-16)$.
4. Multiply: $96 - (-48)$.
5. Add: $96 + 48 = 144$.

Evaluating Scientific Formulae

In university admissions tests, you may encounter complex fractional formulae. Efficiency in calculation is key.

Worked Example: Complex Formula Substitution

A formula is given as $\frac{B}{2} = \frac{P}{3D(D - \sqrt{D^2 - d^2})}$. Find $B$ if $P = 27, D = 17$ and $d = 8$.

First, substitute the values: $\frac{B}{2} = \frac{27}{3(17)(17 - \sqrt{17^2 - 8^2})}$.

Simplify the radical $\sqrt{17^2 - 8^2}$. We can use the difference of two squares to avoid large squaring operations: $17^2 - 8^2 = (17 - 8)(17 + 8) = 9 \times 25 = 225$. Thus, $\sqrt{225} = 15$.

Now, substitute this back into the formula: $\frac{B}{2} = \frac{27}{3(17)(17 - 15)} = \frac{27}{3 \times 17 \times 2}$.

Simplify the fraction by cancelling 27 with 3: $\frac{B}{2} = \frac{9}{17 \times 2}$.

Multiply both sides by 2 to solve for $B$: $B = 2 \times \frac{9}{17 \times 2} = \frac{9}{17}$.