Key Points

Letters like $a$, $x$, or $n$ are used to represent unknown or changing numbers. These are called letter-numbers or variables, and they help us write general rules and formulas.

An algebraic expression is a combination of numbers and letter-numbers (variables) connected by mathematical operations like addition, subtraction, multiplication, and division. For example, $a+3$ and $2y-5$ are algebraic expressions.

To create an expression, translate a verbal phrase into a mathematical statement. For instance, '5 more than a number $x$' becomes $x+5$, and '4 less than 2 times a number $y$' becomes $2y-4$.

To evaluate an expression, substitute the given numerical value for the letter-number and then perform the calculation. If $m=2$ in the expression $5m+3$, its value is $5 imes 2 + 3 = 13$.

The multiplication sign ($imes$) is often omitted between a number and a letter-number. For instance, $4 imes n$ is written as $4n$. This makes expressions more concise.

Like terms are terms that have the exact same letter-numbers. Their numerical coefficients can be different. For example, $5c$, $3c$, and $-c$ are all like terms.

Unlike terms have different letter-numbers. For example, $18c$ and $11d$ are unlike terms because their variables ($c$ and $d$) are different.

You can only add or subtract like terms. To combine them, add or subtract their numerical coefficients and keep the letter-number part the same. For example, $5c + 3c = (5+3)c = 8c$.

Unlike terms cannot be combined into a single term. An expression like $18c + 11d$ is already in its simplest form because $c$ and $d$ are different variables.

The distributive property helps multiply a term by an expression in brackets. For example, $4(x+y)$ is simplified to $4x+4y$, and $3(a-2b)$ becomes $3a-6b$.

When removing brackets that have a negative sign in front, you must change the sign of every term inside the brackets. For example, $-(6x+10y)$ becomes $-6x-10y$.

To simplify an expression, first remove any brackets. Then, group all the like terms together and combine them. For instance, $(7p-3q) + (8p-4q) = 7p+8p-3q-4q = 15p-7q$.

Algebraic expressions are used to create formulas. The perimeter of a rectangle with length $l$ and breadth $b$ is given by the expression $p = l+b+l+b$, which simplifies to $p = 2l+2b$.

A general rule for a pattern can be written as an algebraic expression. For a matchstick pattern where each step adds 2 sticks to a base of 1, the number of sticks for step $y$ is $2y+1$.

Be careful not to confuse different operations. The expression $5u$ means $5 imes u$, while $5+u$ means 5 is added to $u$. These two expressions are not the same and give different values.