Algebra Foundations: cheat sheet

formula

A rule that gives one quantity from others, written with letters. This one gives the perimeter of a rectangle.

P = 2l + 2w

more than, less than

"5 more than n" is $n + 5$ . "5 less than n" is $n - 5$ : you start from $n$ and take 5 away, so $n$ comes first.

n + 5 \qquad n - 5

2x means 2 times x

With $x = 4$ , $2x$ is 8, not 24.

2x = 2 \times x

substitute

Replace each letter with its value, write in the × signs, then work it out: brackets, then × and ÷, then + and −.

\begin{aligned} P &= 2l + 2w \\ &= 2 \times 5 + 2 \times 3 \\ &= 16 \end{aligned}

variable

A letter for a quantity that can change. Each rectangle has its own $l$ , $w$ and $P$ .

P, \; l, \; w

constant

A number that stays the same, such as a 20 dollar call-out fee. $\pi$ is a constant too.

C = 5h + \mathbf{20}

unknown

In an equation, a letter can stand for one particular number: here only $x = 4$ works.

3x + 2 = 14

brackets

Brackets keep a part together so it is worked out first. "Add 3, then double the result" is $2(n + 3)$ ; "double, then add 3" is $2n + 3$ .

2(n + 3) \ne 2n + 3

write an expression for a situation

Say what the letter stands for (a number that changes), multiply it by the amount for each one, then add anything paid once.

\begin{aligned} &\text{Let } g = \text{number of guests} \\ &C = 5g + 60 \end{aligned}

find an unknown in a formula

Substitute what you know, then solve the equation that is left by undoing. Check by substituting back.

\begin{aligned} 20 &= 2 \times 6 + 2w \\ 8 &= 2w \\ w &= 4 \end{aligned}

expression

A calculation with a letter in it. It has a value once you know the letter, but it does not say that anything is equal.

2x + 3

solution

The value of the letter that makes the equation true.

x = 4

inverse operations

Each step has an opposite that undoes it: adding and subtracting, multiplying and dividing.

+ \leftrightarrow -\qquad \times \leftrightarrow \div

backtracking

Undo the steps done to $x$ with their opposites, last step first.

\begin{aligned} 3x + 7 &= 19 \\ 19 - 7 &= 12 \\ 12 \div 3 &= 4 \end{aligned}

Multiply by 3, then add 7. Undo + 7 first.

x on both sides

Take the same number of $x$ from both sides first, so the bags are on one side only.

\begin{aligned} 5x &= 2x + 12 \\ 3x &= 12 \\ x &= 4 \end{aligned}

equation

Two expressions with = between them: a statement that they are equal, like a balanced scale.

2x + 3 = 11

check by substitution

Put the value in place of the letter and work out both sides. If they match, it is a solution. Check every answer in the first equation.

2(4) + 3 = 11 \checkmark

balance method

Do the same to both sides (take away, or share into equal groups) until $x$ is on its own.

\begin{aligned} 3x + 5 &= 20 \\ 3x &= 15 \\ x &= 5 \end{aligned}

Take 5 from both sides, then share both into 3 groups.

expand

Multiply every term inside the bracket by the number outside. Signs come too: $4(3x - 2) = 12x - 8$ .

3(2x + 4) = 6x + 12

change the subject

Undo what was done to the letter, last step first, doing the same to both sides. Check by putting in numbers: $x = 2$ gives $y = 11$ , and $(11 - 5) \div 3 = 2$ .

y = 3x + 5 so x = (y - 5)/3

like terms

Terms with the same letters and powers can be added: $6x - x = 5x$ . Collect them after expanding.

6x + 12 - x = 5x + 12

negative outside

A negative outside flips the sign of every term inside: $-2 \times (-5) = +10$ .

-2(x - 5) = -2x + 10

factorise

Divide every term by the highest common factor and put it outside: $12x - 8 = 4(3x - 2)$ , and $2x^2 + 4x = 2x(x + 2)$ . Check by expanding.

6x + 9 = 3(2x + 3)

6 x-tiles and 9 ones shared into 3 rows of 2x + 3

x on both sides

Expand any brackets. Take the smaller $x$ term from both sides, undo the numbers with the inverse (add to undo $-$ ), then divide.

3x + 7 = 5x - 9 \;\Rightarrow\; 16 = 2x \;\Rightarrow\; x = 8

check by substituting

Put the answer into both sides of the first equation. Equal sides: right. Unequal: look for the first step that went wrong.

3(8) + 7 = 31 = 5(8) - 9

solving an inequality

Do the same to both sides, as for an equation. Multiplying or dividing by a negative **reverses** the sign ( $2 < 5$ but $-2 > -5$ ). Check a value from your answer.

-2x > 6 \;\Rightarrow\; x < -3

inequality on a number line

Open circle for $<$ and $>$ (the end is not included), closed for $\le$ and $\ge$ ; shade towards the solutions.

x < 5:\ \circ \text{ at } 5,\ \text{shaded left}

Algebra Foundations: cheat sheet

Expressions

formula

more than, less than

2x means 2 times x

substitute

variable

constant

unknown

brackets

write an expression for a situation

find an unknown in a formula

Equations

expression

solution

inverse operations

backtracking

x on both sides

equation

check by substitution

balance method

Brackets

expand

change the subject

like terms

negative outside

factorise

Equations and inequalities

x on both sides

check by substituting

solving an inequality

inequality on a number line