Introduction To Variables And Expressions
The concept of variables, constants, expressions and equations form the foundation of algebra. A thorough understanding of these early concepts is of utmost importance for a stronger concept in overall mathematics, going forward.
II. Constants & Variables
When a value does not change, we call it a $\underline{constant}$. For example, numerical values like $5, 21, \sqrt{7}$ are examples of constants.
On the other hand, something that does not have a known fixed value is called a $\underline{variable}$. We can use letters from the alphabet like $a, b, c,... x, y, z$ to denote these variables.
Algebraic expressions are formed using one more $\underline{terms}$, where each term can be a constant, variable or a product of a constant and one or more variables.
For example, the term $x$ is a variable term and $17$ is a constant term, while the term $3x$ is a combination of the constant $3$ and the variable $x$. Similarly, if we take the product $11 \times x \times x \times y$ we get the term $11x^2y$.
An expression is formed by a combining one or more terms using the addition operation. For example the expression:
$7x^2 - 42xy + 12$ is a combination of the terms $7x^2$, $(-42xy)$ and $12$, that is $7x^2 + (-42xy) + 12 = 7x^2 - 42xy + 12$
Some more examples of expressions would be:
$3x - 15$ is an expression in a single variable $x$
$5a^3 + 3ab^2 + 14ab$ is an expression with variables $a$ and $b$.
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Now try the question below:
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IV. Terms Of An Expression
The term of an expression is a product of one or more of the following:
- A constant coefficient
- One or more variables, each raised to some power
For example in the expression:
$3x^2y - 25$, the term $3x^2y$ is a product of the coefficient $3$ and $x \times x$ and $y$, that is,
$3x^2y = 3 \times x \times x \times y$
While the constant term $-25$ does not have any variable in it.
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In an expression like terms are the terms which has the same variables including the same exponents.
Therefore $3xy^2$ and $-20xy^2$ are
$\text{like}$
terms, while $3x^2y$ and $3xy^2$ are
$\text{unlike}$
terms.
The order of variables in a term does not matter, as long as they are same, they are still like terms, that is, $4abc$ and $10bac$ are like terms.
Like terms can be added by adding their coefficients (including their signs), while unlike terms cannot be added.
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VI. Degree Of An Expression
To understand the degree of an expression, we will first need to understand the concept of degree of a single term of any expression. Degree of a term is the sum of the powers of all the variables in the term.
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The degree of an expression is the highest degree amongst all the terms in the expression.
VII. Evaluating Expressions
Like we saw before, an algebraic expression consists of one or more variables, each of which can take up multiple values.
When these variables are fixed at some fixed values, then the expression can give us a fixed numeric value. This is how we evaluate an expression.
Let's take an example to understand this:
Let's say a pencil costs $Rs\ 8$ and an eraser costs $Rs\ 3$.
Suppose a man buys $m$ pencils and $n$ erasers, then the total cost will be
$8m + 3n$
For every customer that buys pencils and erasers from this shop, the total cost can be obtained using the above expression.
Suppose Customer A buys $5$ pencils and $3$ erasers, his bill amount can be found by evaluating the expression at $m = 5$ and $n = 3$, which gives us:
$8 \times 5 + 3 \times 3$
Similarly, when customer B buys $7$ pencils and $2$ erasers, the total cost can be found by evaluating the same expression at $m = 7$ and $n = 2$ which gives us:
$8 \times 7 + 3 \times 2 = 62$