Substitution and Algebraic Concepts for the ESAT

Definitions and Vocabulary

In ESAT Mathematics, it is essential to use the correct terminology when describing mathematical relationships. Consider a rectangle with length $l$, width $w$, and perimeter $P$.

A formula is a mathematical rule that relates different variables. For the perimeter of this rectangle, the formula is $P = 2l + 2w$. This establishes a specific relationship between $P$, $l$, and $w$.

An expression is a collection of symbols and numbers that represent a value but do not contain an equals sign. For example, $2l + 2w$ is an expression for the perimeter. Expressions are made up of terms. In $2l + 2w$, there are two terms, $2l$ and $2w$, although an expression can consist of just a single term.

An equation contains an equals sign ($=$) and is true only for specific values of the unknown variable. For instance, $3x + 1 = 7$ is an equation that is only true when $x = 2$.

An identity is a statement that is true for all possible values of the variables involved. This can be indicated using the identity sign ($\equiv$). For example, $6 \equiv 2 + 4$ and $3x + 2x - 4x \equiv x$ are identities because they remain true regardless of what $x$ is.

An inequality describes the relative size of two expressions, using the following symbols:

1. Less than ($<$)
2. Less than or equal to ($\leq$)
3. Greater than ($>$)
4. Greater than or equal to ($\geq$)
5. Not equal to ($\neq$)

Examples include $3 + 4 < 8$, $2x + 6 \leq 3x$, $x^2 > 4$, $4x - 3 \geq -2$, and $6 + 3 \neq 7$.

A factor is a quantity or expression that divides exactly into another quantity or expression without leaving a remainder. For example, $6$ is a factor of $36$, and $xy$ is a factor of $3x^2y^3$. In the expression $x(x + 1)(x + 3)$, the term $(x + 1)$ is a factor.

Factors of Algebraic Expressions

When listing the factors of a product of variables, you must consider all possible combinations.

Example: List the factors of $p^3q^2$.

The factors are every possible combination of $p$ (up to power 3) and $q$ (up to power 2), taken 1, 2, 3, 4, or 5 at a time. The list is: $1, p, q, p^2, pq, q^2, p^3, p^2q, pq^2, p^3q, p^2q^2, p^3q^2$.

Example: List the factors of $x(x + 3)(x - 5)$.

We can view this as $1 \times x \times (x + 3) \times (x - 5)$. By combining these components, we find the factors: $1, x, x + 3, x - 5, x(x + 3), x(x - 5), (x + 3)(x - 5), x(x + 3)(x - 5)$.

Substitution into Algebraic Expressions

Numerical values can be substituted into expressions to evaluate them. To do this correctly, you must follow the order of operations: Brackets, Indices, Division and Multiplication, Addition and Subtraction (BIDMAS).

Example 1: Evaluate $3x^3 - 5(x - y)$ when $x = 2$ and $y = 5$.

Substitute the values: $3 \times 2^3 - 5 \times (2 - 5)$

Apply BIDMAS:

1. Brackets: $3 \times 2^3 - 5 \times -3$
2. Indices: $3 \times 8 - 5 \times -3$
3. Multiplication: $24 - (-15)$
4. Addition/Subtraction: $24 + 15 = 39$

Example 2: Evaluate $6x^2 - 3(y^3 - 2x)$ when $x = 4$ and $y = -2$.

Substitute the values: $6 \times 4^2 - 3 \times ((-2)^3 - 2 \times 4)$

Apply BIDMAS:

1. Brackets: $6 \times 4^2 - 3 \times (-8 - 8)$
2. Indices: $6 \times 16 - 3 \times -16$
3. Multiplication: $96 - (-48)$
4. Addition: $96 + 48 = 144$

Substitution into Formulae

Complex scientific formulae follow the same rules of substitution and BIDMAS.

Example: Calculate $B$ given the formula $\frac{B}{2} = \frac{P}{3D(D - \sqrt{D^2 - d^2})}$ where $P = 27$, $D = 17$, and $d = 8$.

Substitute the known values into the expression: $\frac{B}{2} = \frac{27}{3 \times 17 \times (17 - \sqrt{17^2 - 8^2})}$

Solve the square root part first. Note that $17^2 - 8^2$ can be solved as $(17 - 8)(17 + 8) = 9 \times 25 = 225$. Thus, $\sqrt{225} = 15$. Alternatively, $289 - 64 = 225$.

Continue the calculation: $\frac{B}{2} = \frac{27}{3 \times 17 \times (17 - 15)}$ $\frac{B}{2} = \frac{27}{3 \times 17 \times 2}$

To find $B$, multiply both sides by 2: $B = 2 \times \frac{27}{3 \times 17 \times 2}$ $B = \frac{27}{3 \times 17}$ $B = \frac{9}{17}$