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• The X sign for multiplication is not needed.

\(6 \times k\) is written as \(6k\) (Remember that \(k\) means \(1k\))

• To multiply a term by a number, multiply the numbers

\(2 \times 5a = 10a\)
\(3 \times 2pq = 6pq\)

Remember the rules for multiplying positive and negative numbers:

• When two numbers with the same signs are multiplied the answer is positive

\(6p \times +7 = +42p\)
\(-5 \times -9m = +45m\)

• When two numbers with different signs are multiplied the answer is negative

\(3 \times -7q = -21q\)
\(-4 \times 2t = -8t\)

Example

Simplify \(5 \times 3q\)

Answer

\(15q\)

The expression \( 5 \times 3q\) has been simplified to give \(15q\)

Example

Simplify \(3kj \times 4\)

Answer

\(12kj\)

An answer of \(12jk\) is also correct. The order of the letters is not important, but the number must come first.

Example

Simplify \(-3t \times -4t\)

Solution

Multiply \(-3\) by \(-4\) using the rule for numbers with the same sign\(-3 \times 4 = +12\)\(-3t \times -4t = +12 \times t \times t\)\(= 12t^2\)

Answer

\(12t^2\)

When more than one term is to be multiplied by a number, it is usual to put a bracket around these terms.

For example \(4(3m + n)\) is a shorter way to write \(4 \times 3m + 4 \times n\)Therefore \(4(3m + n) = 12m + 4n\)

The expression \( 4(3m + n)\) has been expanded to give \(12m + 4n\)

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Example

Expand \(3(7p - 2p)\)

Solution

The \(3\) outside the bracket is positive. There is no need to have a \(+\) in front of it.\(3(7p - 2q) = 3 \times 7p = 3 \times -2q\)\(= 21p - 6q\)

Answer

\(3(7p - 2q) = 21p - 6q\)

An expression is made up of a number of terms. It may be possible to simplify an expression by collecting like terms together. Like terms are easy to spot - they have the same letter or letters.

\(y\)       \(3y\)       \(-24y\)      \(487y\)       are all like terms.

\(3y\) and \(3xy\)      are not like terms but      \(3xy\) and \(5yx\)       are.

\(4y\) and \(9y^2\)       are not like terms but      \(4y^2\) and \(9y^2\)      are.

\(-7\)      \(13\)      \(127\)      \(-46\)      \(-10000\)       are like terms.

All numbers are like terms.

Expressions can be simplified by adding / subtracting like terms as required.

Rules for adding and subtracting terms:

• Only like terms can be added / subtracted
• Remember the rules for combining positive and negative numbers and terms
• When numbers have the same sign, keep that sign and add the numbers.

\(7 + 8 = +15\) (or just \(15\)) and \(7t + 8t = +15t\) (or just \(15t\))

\(-7 - 8 = -15\) and \(-7m - 8m = -15m\)

\(-7 - 8 -3 = -18\) and \(-7q - 8q - 3q = -18q\)

• When two numbers have different signs, use the sign in front of the bigger number and subtract the numbers.

\(12 - 10 = 2\) and \(12p - 10p = 2p (or +2p)\)

\(-11 + 8 = -3\) and \(-11k + 8k = -3k\)

\(8 - 9 = -1\) and \(8j - 9j = -j\)

\(-9 + 11 = +2\) and \(-9y + 11y = +2y\)

Example

Simplify \( 5t - 9t\)

Solution

The signs are different (\(5t\) means \(+5t\))

Use the sign in front of the bigger number and subtract the numbers.

\(9\) is bigger than \(5\) so use a \(-\) sign and subtract the numbers

Answer

\(5t - 9t = -4t\)

Example

Simplify \(-5y + 7 - 14y\)

Solution

\(-5y\) and \(-14y\) are like terms so we can combine these. Both terms have a \(-\) sign.

When numbers have the same sign, keep that sign and add the numbers.

\(-5y - 14y = -19y\)

That leaves the term \(+7\) which can be placed before or after the \(-19y\)

Answer

\(-5y + 7 - 14y = -19y + 7\)

or \(-5y + 7 - 14y = 7 - 19y\)

Both answers are equally correct