MU123 Unit 5 Notes

2 Expressions

2.1 What is an expression?

A collection of letters, numbers and/or mathematical symbols arranged in such a way that if numbers are substituted for the letters, then I can work out hte value of the expression.

Letters represent numbers, so the normal rules of arithmetic apply to them. In exactly the same way as they apply to numbers.

In particular BIDMAS rules apply to the letters.

Evaluation is the process of substituting numbers for the letters. Equivalent expresions are those that are same, just written differently

e.g.: $x + x = 2x$

Rearranging is writing an expression in a different way. Simplifying is rearranging in order to make the expression simpler.

Expressions don't contain equals sings. E.g.: $x + x = 2$ is an equation

bicause noe 2 thynges can be moare equalle. -- Robert Recorde, the inventor of $=$

Identities are equations that are true for all values of the variables

$\frac{3}{10}N = 150$ is not identity, it's currect for only one value of $N$.
$x + x = 2x$ is an identity.

unknown is a letter representing a particular number I don't know yet

In $\frac{3}{10}N = 150$ the $N$ is unknown.

variable is a letter that represents any number

In $x + x = 2x$ the $x$ is variable.

2.2 What is a term?

The things that are added or subtracted in an expression. E.g. in following expression: $$ -2xy + 3z - y^2 $$ The $-2xy$, $+3z$, $-y^2$ are terms. Plus / minus signs are part of the term. There's implicit plus in the front of an expression, e.g. $4a + c$ has following terms: $+4a$ and $+c$. When dicussing the terms the plus sign can be omitted alltogether.

An expression is equivalent to the sum of its terms.

Order of terms can be changed since it doesn't matter when adding numbers.

Constant term is a number only, e.g. in $3a - 2$ the term $-2$ is constant. Coefficient of a term is a number in a term of the $\text{a number} \times \text{a combination of letters}$ form, e.g. $2xy$ has coefficient $2$ and $2$ is a term in $xy$.

2.3 Collecting like terms

Like terms are terms that are 'batches of same thing'. Same thing means the letters and the powers of the letters in each term must be the same. E.g. $7\sqrt{A}$ and $3\sqrt{A}$ are like terms. The lower- and upper-case versions of the same letter are different symbols. Terms cancel each other out when the result of collecting those is zero.

3 Simplifying terms

3.1 Simplifying single terms.

Index notation should be used when simplifying a term with samel letters, e.g. $p \times p$ simplifies to $p^2$.

Strategy

Find the overall sign.

Simplify the rest of the coefficient.

Write the letters in alphabetical order, use index notation.

3.2 Simplifying two or more terms

Strategy

Identify the terms. Each term after the first starts with a plus or minus sign that isn't inside brackets.

Simplify each term. Include the sign (plus or minus) at the start of each term.

Collect any like terms

4 Brackets

4.1 Multiplying out brackets

Any expression containing brackets like: $$ 8a + 3b(b - 2a) $$ can be rewritten without brackets by multiplying each term inside the brackets by the multiplier - $3b$ in the example: $$ 8a + 3b \times b - 3b \times 2a $$

Strategy To multiply out brackets in an expression with more than one term

Identify the terms. Each term after the first starts with a plus or minus sign that isn't inside brackets.

Multiply out the brackets in each term. Include the sign (plus or minus) at the start of each resulting term.

Collect any like terms.

Strategy To remove brackets with a plus or minus sign in front

Inf the sign is plus, keep the sign of each term inside the brackets the same.

If the sign is minus, change the sign of each term inside the brackets.

4.2 Algebraic fractions

Are algebraic expressions written using fraction notation, e.g.: $a + \frac{b}{c}$. Fractions are preffered over divisin sign.

Expanding the algebraic fraction is useful technique when there is more terms in th numerator. such as: $$ \frac{2a - 5b + c}{3d} $$ The expresion can be rewritten as multiplication by the reciprocal of the denominator: $$ \frac{1}{3d}(2a - 5b + c) $$ Then I can multiply out the brackets: $$ \frac{2a}{3d} - \frac{5b}{3d} + \frac{c}{3d} $$

Once the algebraic fraction is expanded it may be possible to simplify some of the resulting terms.

5 Linear equations

Particularly linear equations in one unknown

Linear because those equations doesn't include powers, e.g.: $x^2$, thus form straigh lines on a graph.

An equation consists of two expressions:

left-hand side (LHS)
righ-hand side (RHS)

Solving the equation is the process of finding the value of the unknown. Found value is solution - solution satisfies the equation.

An equation can have more than one solution, e.g.: $a^2 = 4$ has two solutions - $a = 2$ and $a = -2$. Also an equation can have no solution at all, e.g.: $a^2 = -1$

5.2 How to solve linear equation

Strategy

do the same thing to both sides

simplify one side or both sides

swap the sides

clear any fraction and multiply out any brackets. To clear a fraction, multiply both sides by a suitable number.

Add or subtract terms on both sides to obtain an equation of the form $\text{a number} \times \text{the unknown} = \text{a number}$