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Algebra

Expressions and Variables

Contents

Introduction

An expression is a mathematical object with various mathematical terms, i.e., variables, coefficients and constants, related with some mathematical operations like addition, subtraction, etc. For example, x, 2x + 5, 4{x^2} - 5y etc. A variable is a symbol without a predetermined value. In an expression, a symbol with a fixed numerical value is referred to as the constant. A term is a part of an expression consisting of a variable, a constant, or both combined through mathematical operations such as multiplication or division, also then the constant is called a coefficient. A coefficient is a quantity that has been multiplied by a variable. Expressions have many uses, including representing real-world issues as well as solving various and complex mathematical equations to determine revenue, cost, etc.

What are expressions and variables?

Expressions are mathematical objects made by an algebraic combination of variables, coefficient, constants and mathematical operations, such as addition, multiplication etc., which are used to represent a value.

E.g., x,\;y + 3,\;5ab,\;m + {n^2} etc. are some of the examples of expressions.

A variable is a symbol that represents an undetermined values in an expression. The English alphabets such as x, y, z etc. are one of the most commonly used symbols to represent the variables.

E.g., In above examples of expressions, x, y, a, b, m and n are the variables. 

Terms in an expression: A term of an expression is an algebraic combination of numbers and symbols which when are used with a mathematical operation result in an expression.

E.g., In above examples of expressions, x is the term in the first example. y and 3 are the terms in the second example. 5ab is the term in the next expression m{\rm{ and }}{n^2} are the terms in the fourth example.

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Types of Expressions and Variables

Types of Expressions

Numeric Expressions

Numeric expressions are those expressions that contain only numbers and represent a numeric value which we can directly find by performing the operations as they are given in the expression. For example, 12 + 13,\;2 \times 5 + 1,\;3(2 - 4) \div 6{\rm{ and }}\frac{3}{5} \cdot 10 - 5

Algebraic Expressions

The expressions that have variables, constants and coefficients combined together by mathematical operations such as addition, multiplication etc. are known as algebraic expressions. Algebraic expressions are divided into various categories based on the number of terms and the degree of the expressions. Degree of an algebraic expression is the highest power of variables in any of the term of the expression.

Types of algebraic expressions based on number of terms:

  1. Monomials: The algebraic expression with only 1 single term is known as monomial. For example: 2x,\;3y,\;5,xyzare some examples of monomials.
  2. Binomials: The algebraic expression with exactly two terms is known as binomial. For example: x + y,\;{x^2} + 2x,\;x - 3{y^2},\;z + 5 are some examples of binomials.
  3. Polynomials: The algebraic expression with more than 2 terms are known as polynomial. For example: x + y + z,\;{x^2} + 2x + 1,\;xy + yz + zx + 5 are some examples of polynomials.

Types of algebraic expressions based on degree:

  1. Linear: The algebraic expressions with degree of 1, i.e., no two variables same or different, are multiplied with each other. For example: x,\;2x + 3y,\;x - 5 are some examples of linear algebraic expressions.
  2. Quadratic: The algebraic expressions with degree of 2, i.e., at least one term has two variables multiplied to each other. For example: {x^2},\;xy,\;{x^2} + 2x + 1,\;{x^2} + 2xy + {y^2} are some examples of quadratic algebraic expressions.

Types of Variables

Independent Variables

The variables whose value does not depend on any other variable are known as independent variables. Changing other variables does not affect the dependent variables in any way.

Dependent Variables

The variables whose value depends on at least one independent variable are known as dependent variable. Changing the independent variables changes the dependent variables as well.

Variables in algebraic expressions are independent until and unless an equality or inequality relation between the expression and another value is introduced.

For example the variables x and y in the expression x + {y^2} are independent, but if we write the expression as the equation, x + {y^2} = 0 the variables become dependent, since changing the value of one changes the other. We can also write the expression in terms of a different variable like, z = x + {y^2}, then x and y remain independent but z is a dependent variable.

Evaluating Expressions and Solving Equations

Evaluating expressions means that we find the value of an expression by performing the given operations at a certain given values of variables (in case of algebraic expressions). To evaluate the given expression, we follow the BODMAS rule of operations.

The BODMAS rule states that the order of operations in an expression, be it numeric or algebraic, has a certain hierarchy and to find the correct value of the expression we need to follow it properly. The order of operations goes as follows

Bracket – Order – Division – Multiplication – Addition – Subtraction

Hence, comes the name BODMAS, firstly Bracket, the name explains it all if in an expression brackets or parenthesis are given then we start the calculation from inside the parenthesis ignoring the rest until the bracket is completely solved. Next is Order, order is another name for exponents, thus the next step in solving the expression is solving whatever exponent terms are given in the expression so as to reduce every term to a linear term. Then we have the 4 basic operations, i.e., Division, Multiplication, Addition and Subtraction which have to be performed in that specific order.

Expressions can also be equated to some number and then we can find the value of the variables that when substituted in, satisfy the equality. This is known as solving equations.

We have certain conditions for solving equations, to find a unique solution the number of variables must not be greater than the number of equations, thus if we have an equation then we can only solve it for a unique solution if the number of variables in that equation is also 1, if there are 2 or more variables in 1 single equation then we can only guess the possible values of the variables that satisfy the said equation.

Examples of Expressions and Variables

Let’s see how can we solve a numeric expression.

Q. Evaluate \frac{2}{3} \times {(2 \cdot 3 - 3)^2} + 5

By BODMAS rule, we will first simplify the expression inside the parenthesis, then moving onto exponent and then the rest of the operations in order Division, Multiplication, Subtraction and Addition.

    \[\begin{array}{l} \Rightarrow \frac{2}{3} \times {(6 - 3)^2} + 5\\ \Rightarrow \frac{2}{3} \times {(3)^2} + 5\\ \Rightarrow \frac{2}{3} \times 9 + 5\\ \Rightarrow 2 \times 3 + 5\\ \Rightarrow 6 + 5\\ \Rightarrow 11\end{array}\]

Hence, the value of the given numeric expression is 11.

Now, that we have seen how to solve numeric expressions, we can also see the evaluation of algebraic expressions at given values of variables.

Q. Evaluate 3{x^3} + 2xy(x + 2y) - 5{y^3} at x = 2{\rm{ and }}y = 3

First, we will simplify the expressions using distributive law, so as to get separate terms.

Thus, we have the expression,

    \[3{x^3} + 2{x^2}y + 4x{y^2} - 5{y^3}\]

Substituting x = 2{\rm{ and }}y = 3,

    \[\begin{array}{l} \Rightarrow 3{(2)^3} + 2{(2)^2}(3) + 4(2){(3)^2} - 5{(3)^2}\\ \Rightarrow 3(8) + 2(4)(3) + 4(2)(9) - 5(27)\\ \Rightarrow 24 + 24 + 72 - 135\\ \Rightarrow 120 - 135\\ \Rightarrow - 15\end{array}\]

Thus, the value of the given expression at x = 2{\rm{ and }}y = 3 is -15.

Let’s see how can we find the value of a variable from an equation of an expression in that variable,

Q. Solve for x, \frac{5}{2}x - 15 = 10

We will find the value of the said variable by separating the variable from all the numeric values by performing inverse operations, such as if a term is in addition to the term with variable we will subtract that term from both sides,

    \[\frac{5}{2}x - 15 = 10\]

15 is subtracted from the term with x, so we will add 15 to the both sides,

    \[\begin{array}{l} \Rightarrow \frac{5}{2}x - 15 + 15 = 10 + 15\\ \Rightarrow \frac{5}{2}x = 25\end{array}\]

Now, the term with x is divided by 2, so we will multiply by 2 on both sides,

    \[\begin{array}{l} \Rightarrow \frac{5}{2}x \times 2 = 25 \times 2\\ \Rightarrow 5x = 50\end{array}\]

Lastly, x is multiplied by 5, so we will divide both sides by 5,

    \[\begin{array}{l} \Rightarrow \frac{{5x}}{5} = \frac{{50}}{5}\\ \Rightarrow x = 10\end{array}\]

Thus, on solving the equation we have found that x = 10 satisfies the equation.

Conclusion

In this article, we have learned about expressions and variables. Expressions are the mathematical object formed by a combination of variables, coefficients and constants. Variables are the symbols that represent the undetermined value in an expression. In life we come across situations in which we need to calculate some data about multiple things each having different quantities and qualities, we can sometimes represent such data mathematically using expressions in certain variables and coefficients, and then solving the expression can give us the desired results. When solving expressions, we need to make sure that the order of operations we use should be in accordance with BODMAS.

Solved Examples

Example 1: Simplify the following

    \[2(5 - 2) + 3/3 - 5\]

Solution:

We will simplify the given expression using BODMAS,

    \[\begin{array}{l} \Rightarrow 2(3) + 3/3 - 5\\ \Rightarrow 6 + 1 - 5\\ \Rightarrow 2\end{array}\]

Thus, the value of the given expression is 2.

Example 2: Evaluate the following expression for x = \frac{3}{2}{\rm{ and }}y = 2

    \[8{x^2} - 3xy + 5{y^2}\]

Solution: We have, 

    \[8{x^2} - 3xy + 5{y^2}\]

Since, the terms are separate, we will substitute the values of x = \frac{3}{2}{\rm{ and }}y = 2 and simplify, 

    \[\begin{array}{l} \Rightarrow 8{\left( {\frac{3}{2}} \right)^2} - 3\left( {\frac{3}{2}} \right)(2) + 5{(2)^2}\\ \Rightarrow 8\left( {\frac{9}{4}} \right) - 3\left( {\frac{3}{2}} \right)(2) + 5(4)\\ \Rightarrow 2(9) - 3(3) + 5(4)\\ \Rightarrow 18 - 9 + 20\\ \Rightarrow 29\end{array}\]

Thus, the value of the given expression at the given values of the variable is 29.

Example 3: Solve the following equation for p

    \[2p - 3 = \frac{5}{3}p - 2\]

Solution:

We will separate the terms with the variables and the constant terms on either side of the equation using inverse operations,

Adding 3 on both sides,

    \[\begin{array}{l} \Rightarrow 2p - 3 + 3 = \frac{5}{3}p - 2 + 3\\ \Rightarrow 2p = \frac{5}{3}p + 1\end{array}\]

Now since, we have the term, \frac{5}{3}p on the right side, we will subtract both sides by it so as to move it to the left side, 

\begin{array}{l} \Rightarrow 2p - \frac{5}{3}p = \frac{5}{3}p + 1 - \frac{5}{3}p\\ \Rightarrow \frac{{6p - 5p}}{3} = 1\\ \Rightarrow \frac{1}{3}p = 1\end{array}

Now we can multiply 3 on both sided and we will have p on one side and its value on the other,

    \[\begin{array}{l} \Rightarrow \frac{1}{3}p \times 3 = 1 \times 3\\ \Rightarrow p = 3\end{array}\]

Thus, p = 3 is a solution of the given equation.

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FAQs

1. What do non-linear expressions mean?

Ans: Non-linear expressions, as the name suggests, are those expressions where the degree, i.e., the highest power of any variable terms, is more than 1. The non-linear expression with degree 2 has a special name, i.e., quadratic expression.

2. What is the significance of expressions?

Ans: Expressions are the mathematical objects in variables and numbers combined using mathematical operators. Since the variables are undetermined values represented by a symbol, we can assign a variable to any quantity from any real-life scenario, and then solve the expression to find that undetermined quantity.

3. How can you solve a pair of linear equations in 2 variables?

Ans: To solve a pair of linear equations in 2 variables, we use the two equations to eliminate one of the variables from any one equation and then we can find the value of the remaining variable in that equation. Once we have found the value of one of the variables, we can substitute that value in any of the two equations and find the value of the second variable as well.

4. Where in life can we see the use of expressions?

Ans: Expressions are seen in almost every branch of science; we represent various data using variables and their relations give us expressions. For example, if a company sells xarticles in a year at a price of 10 per article, and the production cost is7 per article, excluding

    10000 charges per year for miscellaneous expenses, then the profit can be represented by, <img src="https://wiingy.com/assets/wp-content/ql-cache/quicklatex.com-5bbef15603606928cdb38120cc1a58c5_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="P(x) = 10x - 7x - 10000" title="Rendered by QuickLaTeX.com" height="19" width="197" style="vertical-align: -5px;"/>. Now we can easily find the companies' profit if we know how many articles were sold and inversely, we can also find the number of articles that must be sold to make a certain profit. <!-- /wp:paragraph --> <!-- wp:paragraph --> <strong>5. What does dependent and independent variables have in common?</strong> <!-- /wp:paragraph --> <!-- wp:paragraph --> <strong>Ans: </strong>Dependent and independent variables are undetermined quantities. Thus, unless we are certain which variable represents what object/quantity we cannot be certain which is which, such as if we have,<span class="ql-right-eqno"> </span><span class="ql-left-eqno"> </span><img src="https://wiingy.com/assets/wp-content/ql-cache/quicklatex.com-6a568f0229fcf5383bcaa9b7c0bafc87_l3.png" height="21" width="81" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[z = x + {y^2}\]" title="Rendered by QuickLaTeX.com"/>Then by this representation, we have z as the dependent variable and x and y as independent. But we can rewrite the equation as,<span class="ql-right-eqno"> </span><span class="ql-left-eqno"> </span><img src="https://wiingy.com/assets/wp-content/ql-cache/quicklatex.com-a24a82e9a9db16cb22e86990837b0e4c_l3.png" height="21" width="81" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[x = z - {y^2}\]" title="Rendered by QuickLaTeX.com"/>And now x is the dependent variable whereas z and y are the independent variables. <!-- /wp:paragraph --> <!-- wp:html --> <script type="application/ld+json"> { "@context": "https://schema.org", "@type": "FAQPage", "mainEntity": [{ "@type": "Question", "name": "What do non-linear expressions mean?", "acceptedAnswer": { "@type": "Answer", "text": "Non-linear expressions, as the name suggests, are those expressions where the degree, i.e., the highest power of any variable terms, is more than 1. The non-linear expression with degree 2 has a special name, i.e., quadratic expression." } },{ "@type": "Question", "name": "What is the significance of expressions?", "acceptedAnswer": { "@type": "Answer", "text": "Expressions are the mathematical objects in variables and numbers combined using mathematical operators. Since the variables are undetermined values represented by a symbol, we can assign a variable to any quantity from any real-life scenario, and then solve the expression to find that undetermined quantity." } },{ "@type": "Question", "name": "How can you solve a pair of linear equations in 2 variables?", "acceptedAnswer": { "@type": "Answer", "text": "To solve a pair of linear equations in 2 variables, we use the two equations to eliminate one of the variables from any one equation and then we can find the value of the remaining variable in that equation. Once we have found the value of one of the variables, we can substitute that value in any of the two equations and find the value of the second variable as well." } },{ "@type": "Question", "name": "Where in life can we see the use of expressions?", "acceptedAnswer": { "@type": "Answer", "text": "Expressions are seen in almost every branch of science; we represent various data using variables and their relations give us expressions. For example, if a company sells <img src="https://wiingy.com/assets/wp-content/ql-cache/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="x" title="Rendered by QuickLaTeX.com" height="8" width="10" style="vertical-align: 0px;"/>articles in a year at a price of

10 per article, and the production cost is 7 per article, excluding10000 charges per year for miscellaneous expenses, then the profit can be represented by, P(x) = 10x - 7x - 10000. Now we can easily find the companies’ profit if we know how many articles were sold and inversely, we can also find the number of articles that must be sold to make a certain profit.” } },{ “@type”: “Question”, “name”: “What does dependent and independent variables have in common?”, “acceptedAnswer”: { “@type”: “Answer”, “text”: “Dependent and independent variables are undetermined quantities. Thus, unless we are certain which variable represents what object/quantity we cannot be certain which is which, such as if we have,

    \[z = x + {y^2}\]

Then by this representation, we have z as the dependent variable and x and y as independent. But we can rewrite the equation as,

    \[x = z - {y^2}\]

And now x is the dependent variable whereas z and y are the independent variables.” } }] }

References

https://byjus.com/maths/algebraic-expressions/

https://www.cuemath.com/algebra/algebraic-expression/

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