Algebraic Manipulation and Equations for ESAT Mathematics 1

Algebraic Notation and Index Laws

Algebraic notation allows us to express mathematical relationships concisely. Standard conventions include writing $ab$ instead of $a \times b$, $3y$ instead of $y + y + y$, and $a/b$ instead of $a \div b$. When combining letters and numbers into terms, numerical coefficients always come first, followed by letters in alphabetical order, such as $6pqr$.

Index laws govern the multiplication and division of powers. To multiply powers with the same base, we add the indices: $a^m \times a^n = a^{m+n}$. To divide them, we subtract the indices: $a^m \div a^n = a^{m-n}$. When raising a term to a further power, we multiply the indices: $(a^m)^n = a^{mn}$. Note that any non zero base raised to the power of zero is $1$, and $a^1$ is simply $a$. For negative powers, $a^{-m} = 1/a^m$. Fractional powers represent roots, where $a^{1/b} = \sqrt[b]{a}$ and $a^{m/n} = \sqrt[n]{a^m}$.

Substitution and Definitions

Understanding vocabulary is vital for interpreting ESAT questions. An expression like $2l + 2w$ does not have an equals sign, whereas an equation like $3x + 1 = 7$ is true only for specific values of the variable. An identity, denoted by $\equiv$, is true for all values of the variable, such as $3x + 2x \equiv 5x$. An inequality describes the relative size of expressions using symbols like $<, >, \leq, \geq,$ and $\neq$. A factor is a quantity that divides exactly into another without a remainder.

When substituting numerical values into formulae, the BIDMAS order of operations must be followed. For example, if $x = 4$ and $y = -2$ in the expression $6x^2 - 3(y^3 - 2x)$, we evaluate indices and brackets first: $6(16) - 3(-8 - 8) = 96 - 3(-16) = 96 + 48 = 144$.

Expanding and Factorising Expressions

Like terms are terms that are identical except for their numerical coefficients, such as $12x^2y^4$ and $-6x^2y^4$. These can be collected by adding or subtracting. Expanding involves multiplying a single term over a bracket, $a(b + c) = ab + ac$, or expanding binomials using methods like the grid method.

Factorising is the reverse of expanding. For a quadratic of the form $x^2 + bx + c$, we find two numbers that multiply to give $c$ and add to give $b$. For $ax^2 + bx + c$, we can split the middle term. For example, to factorise $6x^2 + 23x + 20$, we find numbers that multiply to $120$ ($6 \times 20$) and add to $23$. These are $15$ and $8$.

$6x^2 + 15x + 8x + 20 = 3x(2x + 5) + 4(2x + 5) = (2x + 5)(3x + 4)$

Special cases include the difference of two squares, where $a^2x^2 - b^2 = (ax + b)(ax - b)$. For example, $16x^2 - 49 = (4x + 7)(4x - 7)$.

Rearranging Formulae and Rational Expressions

Changing the subject of a formula involves isolating a specific variable. You can add, subtract, multiply, or divide both sides by the same quantity, or square/root both sides. For example, to make $x$ the subject of $y = \sqrt{\frac{3x+5}{4}}$, we square to get $y^2 = \frac{3x+5}{4}$, then multiply by $4$ and subtract $5$ to reach $x = \frac{4y^2 - 5}{3}$.

Rational expressions are algebraic fractions. They are simplified by factorising the numerator and denominator and then cancelling common factors. To add or subtract them, find a common denominator, ideally the lowest common multiple (LCM). For example, $\frac{x}{a} + \frac{y}{b} = \frac{xb + ya}{ab}$.

Coordinates and Linear Graphs

Points are located using $(x, y)$ coordinates across four quadrants.

Linear functions follow $y = mx + c$, where $m$ is the gradient and $c$ is the $y$ intercept. Parallel lines share the same gradient. Perpendicular lines have gradients $m_1$ and $m_2$ such that $m_1 \times m_2 = -1$. The gradient between two points is calculated as $\frac{y_2 - y_1}{x_2 - x_1}$.

Quadratic Functions and Their Features

The graph of $y = ax^2 + bx + c$ is a parabola. If $a > 0$, it is U-shaped: if $a < 0$, it is $\cap$ shaped. The roots are where the graph crosses the $x$ axis ($y=0$). The turning point (vertex) is the maximum or minimum point and can be found by completing the square: $x^2 + kx = (x + k/2)^2 - (k/2)^2$.

For $y = x^2 + 6x + 7$, completing the square gives $(x + 3)^2 - 2$. The turning point is where the bracket is zero, at $x = -3, y = -2$.

Other Graph Types

Students must recognise and sketch:

1. Cubic functions: $y = ax^3$ has rotational symmetry about the origin.
2. Reciprocal functions: $y = 1/x$ exists in the first and third quadrants and never touches the axes.
3. Exponential functions: $y = k^x$ always passes through $(0, 1)$. If $k > 1$, it shows rapid growth: if $0 < k < 1$, it shows decay.
4. Trigonometric functions: $\sin x$ and $\cos x$ oscillate between $1$ and $-1$, while $\tan x$ has periodic asymptotes.

Graphs in Context

Graphs can model real world situations. In a distance time graph, the gradient is the speed. In a speed time graph, the gradient is acceleration and the area under the graph is the total distance travelled.

To find the area under a curve, we divide the region into strips, often trapezia, and sum their areas. The gradient of a curve at any point is found by drawing a tangent and calculating its slope.

Solving Equations and Simultaneous Systems

Simultaneous equations can be solved algebraically or graphically. To solve graphically, plot both lines and find the intersection. Algebraically, use elimination or substitution. For a system with one linear and one quadratic equation, always use substitution. For example, if $x + 4y = 7$ and $x^2 + 4xy + 2y^2 = 1$, substitute $x = 7 - 4y$ into the quadratic equation to solve for $y$, then find $x$.

Quadratic equations are solved by factorising, completing the square, or the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Linear Inequalities

Inequalities are solved similarly to equations, but multiplying or dividing by a negative number reverses the sign. Solutions are represented on number lines: open circles for $<$ or $>$ and solid circles for $\leq$ or $\geq$. On graphs, regions are defined by boundary lines, often shading out the unwanted area.

Sequences

Sequences can be defined by term to term rules (how to get the next number) or position to term rules (the $n$th term).

A linear sequence has a constant first difference. If the difference is $d$, the $n$th term is $dn + c$. A quadratic sequence has a constant second difference. The $n$th term is $an^2 + bn + c$, where $a$ is half the second difference. For example, the sequence $5, 14, 27, 44...$ has a constant second difference of $4$, so $a = 2$. By comparing with $2n^2$, we find the $n$th term is $2n^2 + 3n$.