Expressions and Formulas

Growing up as a teenager, I never really liked WORD PROBLEMS, not because they were difficult, but because of the name WORD PROBLEM. Couldn't it have been named word-task, word-maths, or word-anything but not PROBLEM because the word, PROBLEM, in itself was psychologically a problem to me? However, not long after, I began to understand that these PROBLEMS are indeed best tackled when expressed in algebra, equations, and formulas.

Expressions and Formulas Expressions and Formulas

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    Hence hereafter, you would be learning about the evaluation of algebraic expressions and formulas, their rules and steps along with some fascinating examples to juggle your memory. Go EXPRESSION, no PROBLEM!

    Evaluating Algebraic expressions with examples

    Expressions are illustrations of numbers, variables, and terms that are combined by operations such as addition, subtraction, division, and multiplication. In other words, these are sentences used by mathematicians in communicating their ideas.

    Dealing with algebraic expressions, letters are mostly used to represent numbers. These letters are called variables. The combination of these letters with numbers and mathematical operations is what is called an algebraic expression. Using \(x\) as a variable, we can take a look at a few examples of algebraic expressions.

    1. \(2x + 1\)
    2. \(23 - 5 + x^2\)
    3. \(x^2+6x+(−3)\)

    How would a mathematician express this detail, "In \(5\) years' time a father would be twice the current age of his son"?

    Solution:

    Let the father's current age be \(f\) and that of his son be \(s\). In five years time the father's age would be:

    \[f+5\]

    But we were told that his age would be twice that of his son by that time, so, this can be expressed as:

    \[2s\]

    If this were to be combined to make sense it becomes:

    \[f+5=2s\]

    That is the way a mathematician would represent that information.

    Variables, terms, and coefficients

    Variables are the letter components of expressions. They are ones that distinguish expressions from arithmetic operations. Terms are the components of expressions that are separated by addition or subtraction. Coefficients are the numerical factors multiplying variables.

    For example, if we had the expression \(x^2y^2+6xy+(-3)\), we could identify \(x\) and \(y\) as the variable components of the expression. The number \(6\) is identified as the coefficient of the term \(6xy\). The number \(-3\) is called a constant. The identified terms here are \(x^2y^2\), \(6xy\) and \(-3\).

    We can take a few examples and categorize their components under either variables, coefficients, or terms.

    1. \(\frac{4}{5}y+14x-3\)
    2. \(2^2-4x\)
    3. \(\frac{1}{2}+x^2y^3\)

    Expressions

    Variables

    Coefficients

    Constants

    Terms

    \(\frac{4}{5}y+14x-3\)

    \(x\) and \(y\)

    \(\frac{4}{5}\) and \(14\)

    \(-3\)

    \(\frac{4}{5}y\), \(14x\) and \(-3\)

    \(2^2-4x\)

    \(x\)

    \(-4\)

    \(2^2\)

    \(2^2\) and \(-4x\)

    \(\frac{1}{2}+x^2y^3\)

    \(x\) and \(y\)

    \(1\) (though it's not shown, this is technically the coefficient of \(x\) and \(y\))

    \(\frac{1}{2}\)

    \(\frac{1}{2}\), \(x^2y^3\)

    Rules for evaluating expressions

    Rules which are followed when evaluating expressions are otherwise referred to as the laws of algebraic expression.

    Rule of commutativity

    The commutative rule applies to both operations of addition and multiplication. It states that the position of the term (number, variable, etc.) which undergoes any of these operations (addition or multiplication) is not a determinant of the result. This means that if you are to find the sum between \(x\) and \(y\) or the product between \(x\) and \(y\) the result will not change even if \(y\) is positioned first, and \(x\) is positioned second in the operation.

    Commutativity means:

    \[x+y=y+x\]

    or

    \[x\times y=y\times x\]

    You should try numbers with this below.

    Prove the commutative law with the following:

    a. \(5+3\)

    b. \(6 \times 11\)

    Solution:

    a. Note that

    \[5+3=8\]

    and

    \[3+5=8\]

    So,

    \[5+3=3+5\]

    Hence, the above expression complies with the commutativity rule.

    b. Note that

    \[6 \times 11=66\]

    and

    \[11 \times 6=66\]

    Hence,

    \[6\times 11=11\times 6\]

    Therefore, the above expression complies with the commutativity rule.

    Note that when the commutativity rule involves addition it is called additive commutativity. Meanwhile, when multiplication is involved, it is called multiplicative commutativity.

    Rule of associativity

    The associative rule applies to both operations of addition and multiplication. It explains that segregating or grouping (by using parenthesis or brackets) does not affect the result of sums or products. This means that if \(x\) and \(y\) were to be segregated and multiplied separately before multiplying with \(z\), it is still equivalent to multiplying a segregate of \(y\) and \(z\) before multiplying with \(x\). This rule also applies to addition.

    Associativity means:

    \[(x+y)+z=x+(y+z)\]

    or

    \[(x\times y)\times z=x\times (y\times z)\]

    You should try numbers with this below.

    Show the associative property using the following:

    a. \((5+3)+7\)

    b. \((3\times 2)\times 4\)

    Solution:

    a. By calculating,

    \[(5+3)+7=8+7=15\]

    and

    \[5+(3+7)=5+10=15\]

    Thus, the above expression obeys the associative rule.

    b. By calculating,

    \[(3\times 2)\times 4=6\times 4=24\]

    and

    \[3\times (2\times 4)=3\times 8=24\]

    In earnest, the above expression shows an associative property.

    Note that when associativity rule involves addition it is called additive associativity. Meanwhile, when multiplication is involved, it is called multiplicative associativity.

    Rule of distributivity

    The distributive law applies when both addition and multiplication operations are carried out simultaneously (at the same time). It states that when a term is multiplied by a segregate (grouped in brackets) operating under addition, the result is equivalent to the sum of the product between that term and the individual components of the segregate. The individual components of the segregate are known as addends or summands. This implies that when a certain term \(x\) is multiplied by a segregate sum of \(y\) and \(z\), it is equivalent to the sum of the products of \(x\) and \(y\) as well as \(x\) and \(z\).

    Distributivity means:

    \[x\times (y+z)=xy+xy\]

    where \(y\) and \(z\) are each called the addend of the sum \((y+z)\).

    You should look at the example below which uses numbers to give you a clearer insight.

    Determine the distributive law using the below expression:

    \[5\times (4+3)\]

    Solution:

    In the first instance,

    \[5\times (4+3)=5\times 7=35\]

    and in the second instance,

    \[5\times (4+3)=(5\times 4)+(5\times 3)\]

    which is

    \[5\times (4+3)=20+15=35\]

    Therefore, the expression \(5\times (4+3)\) complies with the distributivity rule.

    Steps on how to evaluate expressions and formulas

    To evaluate an expression, you need to take two steps.

    1. Substitute each variable's assigned value into the expression. This is best done by using parentheses around each substituted value.

    2. Perform operations on expressions using the Order of Operations.

    Use PEMDAS. Take a look at the Order of Operations article for a good understanding.

    If \(x = 6\), evaluate \(6+x\).

    Solution:

    Substitute \(6\),

    \[6 + 6 \]

    \[12\]

    Evaluate the expression below if \(x = 3\), and \(y = 2\).

    \[6y+4x=?\]

    Solution:

    \[6(2)+4(3)=?\]

    \[12+12=24\]

    Evaluate \[2x^3-x^2+y\]

    if \(x = 3\), and \(y = –2\).

    Solution:

    \[2x^3-x^2+y\]

    \[2(3)^3-(3)^2+(-2)\]

    \[2(27)-9+(-2)\]

    \[54-9+(-2)\]

    \[43\]

    Evaluate \[y^2-y-6\]

    where \(y = –4\).

    Solution:

    \[(-4)^2-(-4)-6\]

    \[16+4-6\]

    \[14\]

    Remember that subtracting a negative is the same as adding a positive.

    Rules of evaluating formulas

    Just as the rules which have been used earlier to describe expressions, formulas are defined by the rules of commutativity, associativity, and distributivity. However, an extra rule is common in evaluating formulas. This is the inverse function rule. This rule helps you to undo what has been done to get express a term of your choice. The table below lists a few of the inverse functions.

    Function (operation)Inverse functionExample
    \(+\)\(-\)\(a+b=c\) then \(a=c-b\)
    \(-\)\(+\)\(x-y=z\) then \(x=z+y\)
    \(\times \)\(\div \)\(a\times b=ab\) then \(ab\div b=a\)
    \(\div \)\(\times \)\(x\div y=z\) then \(x=z\times y\)
    powersroots\(a^n=b\) then \(a=\sqrt[n]{b}\)
    rootspowers\(\sqrt[y]{x}=z\) then \(x=z^y\)

    How to evaluate formulas?

    Evaluating formulas is not so different from how expressions are being evaluated. Formulas contain variables, coefficients, constants, and terms. All that is required is to substitute a value into a formula and operate on them arithmetically. Let us take examples of formulas and how they can be evaluated.

    Find force if the mass of an object is \(20\, \text{kg}\) and the acceleration is \(5\, \text{ms}^{-2}\).

    Solution:

    \[\text{Force}=\text{Mass}\times \text{Acceleration}\]

    We will substitute the numbers into the equation.

    \[\text{Force}=20\times 5\]

    \[\text{Force}=100\, \text{N}\]

    Evaluate the time taken for an object that moved across \(3600\, \text{m}\) in \(2\, \text{ms}^{-1}\).

    Solution:

    \[\text{Time}=\frac{\text{Distance}}{\text{Speed}}\]

    According to the problem,

    \[\text{Distance}=3600\, \text{m}\]

    \[\text{Speed}=2\,\text{ms}^{-1}\]

    Substitute numbers into the formula

    \[\text{Time}=\frac{3600}{2}\]

    \[\text{Time}=1800\, \text{seconds}\]

    Expressions and Formulas - Key takeaways

    • Variables are the letter components of expressions.
    • Terms are the components of expressions that are separated by addition or subtraction.
    • To evaluate an expression, substitute each variable's assigned value into the expression and perform operations on them arithmetically.
    Frequently Asked Questions about Expressions and Formulas

    How to evaluate algebraic expressions and formulas?

    Substitute the values associated with the variables into a formula and operate on them arithmetically.

    What are the rules for evaluating expressions and formulas?

    To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.

    What are examples of expressions and formula evaluation?

    x + 2 + 2x,  where x = 3

    Solution

    3 + 2 + 2(3)

    11

    How to use substitution to evaluate algebraic expressions and formulas.

    Plug in given values for variables in the expression. Use PEMDAS to perform arithmetic operations on them.

    What methods are applied for expression and formula evaluation? 

    The method used in expression and formula evaluation involves two steps such as the substitution of each variable's assigned value into the expression, and performing operations on expressions using the Order of Operations.

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