Sec 1 · G3 (Express)

Basic algebra

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1. Algebraic notation

Write	Means	Example
$3x$	$3 \times x$	$x = 4$ : $3x = 12$
$xy$	$x \times y$
$x^{2}$	$x \times x$	$x = 4$ : $x^{2} = 16$
$\dfrac{x}{5}$	$x \div 5$
$x$	$1x$ (the $1$ is not written)

Words	Expression
the sum of $x$ and $y$	$x + y$
add $5$ to $y$	$y + 5$
$5$ more than $y$	$y + 5$
$y$ increased by $5$	$y + 5$
subtract $5$ from $y$	$y - 5$
$5$ less than $y$	$y - 5$
$y$ decreased by $5$	$y - 5$
$y$ subtracted from $5$	$5 - y$
the product of $x$ and $y$	$xy$
$3$ times $y$	$3y$
$y$ multiplied by $3$	$3y$
$y$ divided by $4$	$\dfrac{y}{4}$
$x$ divided by $y$	$\dfrac{x}{y}$
half of $y$	$\dfrac{y}{2}$
the square of $y$	$y^{2}$

Part of $5x + 3y - 7$	Name
$5x$ , $3y$ , $-7$	terms
$5$ in $5x$	coefficient of $x$
$-7$	constant term

Substitute: put negatives in brackets, x = −3

\displaystyle 2x^{2} - 5x

e.g.

\displaystyle 2(-3)^{2} - 5(-3)

\displaystyle = 18 + 15

\displaystyle = 33

📖 Worked example

Write down an algebraic expression for $8$ less than $y$ .

Step 1 of 2

Which operation do the words describe?

“8 less than y” means start with y and take away 8.

📖 Worked example

Given that $x = -6$ and $y = -5$ , find the value of $x(y + 5)$ .

Step 1 of 3

Substitute the values (negatives and fractions in brackets)

x(y + 5) = \left(-6\right)(-5 + 5)

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2. Simplifying: like terms

Collect the same letters (and the same powers)

\displaystyle 7a + 3b - 2a + 5b

e.g.

\displaystyle 7a + 3b - 2a + 5b

\displaystyle = 5a + 8b

x and x² are not like terms

\displaystyle 4x^{2} + 3x - x^{2} - 5x

e.g.

\displaystyle 4x^{2} + 3x - x^{2} - 5x

\displaystyle = 3x^{2} - 2x

📖 Worked example

Simplify $2x - 6 - 4x - 3$ .

Step 1 of 2

Group the like terms (each term keeps the sign in front of it)

= 2x - 4x - 6 - 3

In $5x^{2} - 3xy + y - 7$
term	a part joined by $+$ or $-$ , with its sign	$5x^{2}$ , $-3xy$ , $y$ , $-7$
coefficient	the number in front of the letters, with its sign	of $xy$ : $-3$ ; of $y$ : $1$
constant term	the term with no letter	$-7$

📖 Worked example

Write down the coefficient of $xy$ in the expression $2 + 3y - 9y^{2} - xy$ .

Step 1 of 2

Find the term in $xy$ : each term keeps the sign in front of it

The term in

xy

-xy

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3. Expanding

Multiply every term inside

\displaystyle a(b + c) = ab + ac

e.g.

\displaystyle 4(2x - 3)

\displaystyle = 8x - 12

A minus outside reaches both terms

\displaystyle -2(3x - 4)

e.g.

\displaystyle -2(3x - 4)

\displaystyle = -6x + 8

Expand, then collect like terms

\displaystyle 3(x + 2) - (x - 5)

e.g.

\displaystyle 3(x + 2) - (x - 5)

\displaystyle = 3x + 6 - x + 5

\displaystyle = 2x + 11

📖 Worked example

Expand and simplify $5(5x + 3) - 3(3x - 3)$ .

Step 1 of 2

Multiply every term in each bracket by the term in front of it (a minus in front changes both signs)

5(5x + 3) - 3(3x - 3) = 25x + 15 - 9x + 9

Words	With brackets
subtract $A$ from $B$	$B - A$
subtract the sum of $A$ and $B$ from $C$	$C - (A + B)$
subtract $A$ from the sum of $B$ and $C$	$B + C - A$

📖 Worked example

$A = 3x + 2x^{2} + 8$ , $B = 5x^{2} - 4 + 2x$ and $C = 3x^{2} - 9$ . Subtract the sum of $A$ and $B$ from $C$ . Give your answer in its simplest form.

Step 1 of 4

Write the words with A, B and C (“subtract X from Y” means Y − X)

C - (A + B)

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4. Factorising

Take out the HCF of every term

\displaystyle 6x + 9y

e.g.

\displaystyle 6x + 9y

\displaystyle = 3(2x + 3y)

The HCF can include a letter

\displaystyle 4ab - 10a

e.g.

\displaystyle 4ab - 10a

\displaystyle = 2a(2b - 5)

📖 Worked example

Factorise $10x - 35$ completely.

Step 1 of 4

Find the HCF of the numbers in front of the letters

\text{HCF of } 10 \text{ and } 35 =

5

A bracket in the question: expand first, then take out the HCF

\displaystyle 6y + 4(3x - 2)

e.g.

\displaystyle 6y + 4(3x - 2)

\displaystyle = 6y + 12x - 8

\displaystyle = 2(3y + 6x - 4)

📖 Worked example

Expand and simplify $11x + 5(x - 8)$ . Then factorise your answer completely.

Step 1 of 5

Expand the bracket first

11x + 5(x - 8) = 11x + 5x - 40

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5. Fractions with letters

Same denominator first

\displaystyle \dfrac{x}{2} + \dfrac{x}{3}

e.g.

\displaystyle \dfrac{x}{2} + \dfrac{x}{3}

\displaystyle = \dfrac{3x}{6} + \dfrac{2x}{6}

\displaystyle = \dfrac{5x}{6}

Bracket the top when subtracting

\displaystyle \dfrac{x + 1}{4} - \dfrac{x - 2}{3}

e.g.

\displaystyle \dfrac{3(x + 1) - 4(x - 2)}{12}

\displaystyle = \dfrac{3x + 3 - 4x + 8}{12}

\displaystyle = \dfrac{11 - x}{12}

📖 Worked example

Express $\dfrac{2x + 3}{5} + \dfrac{x + 5}{2}$ as a single fraction in its simplest form.

Step 1 of 4

Find the LCM of the denominators

\text{LCM of } 5 \text{ and } 2 =

10

Practise this →

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Sec 1 · G3 (Express)

Basic algebra

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1. Algebraic notation

Write	Means	Example
$3x$	$3 \times x$	$x = 4$ : $3x = 12$
$xy$	$x \times y$
$x^{2}$	$x \times x$	$x = 4$ : $x^{2} = 16$
$\dfrac{x}{5}$	$x \div 5$
$x$	$1x$ (the $1$ is not written)

Words	Expression
the sum of $x$ and $y$	$x + y$
add $5$ to $y$	$y + 5$
$5$ more than $y$	$y + 5$
$y$ increased by $5$	$y + 5$
subtract $5$ from $y$	$y - 5$
$5$ less than $y$	$y - 5$
$y$ decreased by $5$	$y - 5$
$y$ subtracted from $5$	$5 - y$
the product of $x$ and $y$	$xy$
$3$ times $y$	$3y$
$y$ multiplied by $3$	$3y$
$y$ divided by $4$	$\dfrac{y}{4}$
$x$ divided by $y$	$\dfrac{x}{y}$
half of $y$	$\dfrac{y}{2}$
the square of $y$	$y^{2}$

Part of $5x + 3y - 7$	Name
$5x$ , $3y$ , $-7$	terms
$5$ in $5x$	coefficient of $x$
$-7$	constant term

Substitute: put negatives in brackets, x = −3

\displaystyle 2x^{2} - 5x

e.g.

\displaystyle 2(-3)^{2} - 5(-3)

\displaystyle = 18 + 15

\displaystyle = 33

📖 Worked example

Write down an algebraic expression for $8$ less than $y$ .

Step 1 of 2

Which operation do the words describe?

“8 less than y” means start with y and take away 8.

📖 Worked example

Given that $x = -6$ and $y = -5$ , find the value of $x(y + 5)$ .

Step 1 of 3

Substitute the values (negatives and fractions in brackets)

x(y + 5) = \left(-6\right)(-5 + 5)

Practise this →

2. Simplifying: like terms

Collect the same letters (and the same powers)

\displaystyle 7a + 3b - 2a + 5b

e.g.

\displaystyle 7a + 3b - 2a + 5b

\displaystyle = 5a + 8b

x and x² are not like terms

\displaystyle 4x^{2} + 3x - x^{2} - 5x

e.g.

\displaystyle 4x^{2} + 3x - x^{2} - 5x

\displaystyle = 3x^{2} - 2x

📖 Worked example

Simplify $2x - 6 - 4x - 3$ .

Step 1 of 2

Group the like terms (each term keeps the sign in front of it)

= 2x - 4x - 6 - 3

In $5x^{2} - 3xy + y - 7$
term	a part joined by $+$ or $-$ , with its sign	$5x^{2}$ , $-3xy$ , $y$ , $-7$
coefficient	the number in front of the letters, with its sign	of $xy$ : $-3$ ; of $y$ : $1$
constant term	the term with no letter	$-7$

📖 Worked example

Write down the coefficient of $xy$ in the expression $2 + 3y - 9y^{2} - xy$ .

Step 1 of 2

Find the term in $xy$ : each term keeps the sign in front of it

The term in

xy

-xy

Practise this →

3. Expanding

Multiply every term inside

\displaystyle a(b + c) = ab + ac

e.g.

\displaystyle 4(2x - 3)

\displaystyle = 8x - 12

A minus outside reaches both terms

\displaystyle -2(3x - 4)

e.g.

\displaystyle -2(3x - 4)

\displaystyle = -6x + 8

Expand, then collect like terms

\displaystyle 3(x + 2) - (x - 5)

e.g.

\displaystyle 3(x + 2) - (x - 5)

\displaystyle = 3x + 6 - x + 5

\displaystyle = 2x + 11

📖 Worked example

Expand and simplify $5(5x + 3) - 3(3x - 3)$ .

Step 1 of 2

Multiply every term in each bracket by the term in front of it (a minus in front changes both signs)

5(5x + 3) - 3(3x - 3) = 25x + 15 - 9x + 9

Words	With brackets
subtract $A$ from $B$	$B - A$
subtract the sum of $A$ and $B$ from $C$	$C - (A + B)$
subtract $A$ from the sum of $B$ and $C$	$B + C - A$

📖 Worked example

$A = 3x + 2x^{2} + 8$ , $B = 5x^{2} - 4 + 2x$ and $C = 3x^{2} - 9$ . Subtract the sum of $A$ and $B$ from $C$ . Give your answer in its simplest form.

Step 1 of 4

Write the words with A, B and C (“subtract X from Y” means Y − X)

C - (A + B)

Practise this →

4. Factorising

Take out the HCF of every term

\displaystyle 6x + 9y

e.g.

\displaystyle 6x + 9y

\displaystyle = 3(2x + 3y)

The HCF can include a letter

\displaystyle 4ab - 10a

e.g.

\displaystyle 4ab - 10a

\displaystyle = 2a(2b - 5)

📖 Worked example

Factorise $10x - 35$ completely.

Step 1 of 4

Find the HCF of the numbers in front of the letters

\text{HCF of } 10 \text{ and } 35 =

5

A bracket in the question: expand first, then take out the HCF

\displaystyle 6y + 4(3x - 2)

e.g.

\displaystyle 6y + 4(3x - 2)

\displaystyle = 6y + 12x - 8

\displaystyle = 2(3y + 6x - 4)

📖 Worked example

Expand and simplify $11x + 5(x - 8)$ . Then factorise your answer completely.

Step 1 of 5

Expand the bracket first

11x + 5(x - 8) = 11x + 5x - 40

Practise this →

5. Fractions with letters

Same denominator first

\displaystyle \dfrac{x}{2} + \dfrac{x}{3}

e.g.

\displaystyle \dfrac{x}{2} + \dfrac{x}{3}

\displaystyle = \dfrac{3x}{6} + \dfrac{2x}{6}

\displaystyle = \dfrac{5x}{6}

Bracket the top when subtracting

\displaystyle \dfrac{x + 1}{4} - \dfrac{x - 2}{3}

e.g.

\displaystyle \dfrac{3(x + 1) - 4(x - 2)}{12}

\displaystyle = \dfrac{3x + 3 - 4x + 8}{12}

\displaystyle = \dfrac{11 - x}{12}

📖 Worked example

Express $\dfrac{2x + 3}{5} + \dfrac{x + 5}{2}$ as a single fraction in its simplest form.

Step 1 of 4

Find the LCM of the denominators

\text{LCM of } 5 \text{ and } 2 =

10

Practise this →

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