What comes to your mind when you think about variables? Maybe something that changes like someone's height, weight, age, or even their grades in an exam? Yes, you are on the right track. In Algebra, we use letters as variables to represent unknown values. Why are they unknown, you ask? Well, it is not because they are mysterious numbers, but because they can take many values. Using variables allows us to represent and solve mathematical problems with unknown values, and also to generalize mathematical concepts.
In this article, we will define what variables are in Algebra, and how to identify and work with variables in algebraic expressions and equations. We will also explore the Order of Operations with variables, and the concepts of dependent and independent variables, showing you practical examples along the way.
Let's start by looking into the origin of variables.
Introduction to Variables in Algebra
Sometimes when you need to solve mathematical problems, you might encounter unknown values, which are values that can change. In the 17th century, René Descartes came up with a great idea, which was to represent unknowns in equations by \(x\), \(y\), and \(z\), and knowns by \(a\), \(b\), and \(c\).1 The idea was to use letters as if they were numbers, to be able to work out general solutions to problems, where not all values are known. This made things easier for Mathematicians, saving them time and effort in finding every possible solution.
What are Variables in Algebra?
In Algebra, variables are letters that are used to represent unknown values that can change.
Let's see some examples in the section below.
Examples of Variables in Algebra
The most common letters that are used as variables in Algebra are \(x, y, z, a, b, m, n, p,\) and \(q\). From these, the most popular ones are \(x\) and \(y\). In science, \(t\) and \(d\) are very common variables used when referring to time and distance, respectively.
Here are a few examples of variables:
a) You can define a variable \(h\) as the numbers of hours that you spend on the Internet per day.
\[h = \text{number of hours spent on the Internet per day}\]
b) A variable \(m\) as the number of milk cartons sold per day in your local shop.
\[m = \text{number of milk cartons sold per day}\]
c) A variable \(d\) as the number of days until your next holiday.
\[d = \text{number of days until holiday}\]
Variables in Algebraic Expressions
Variables are of utmost importance in Algebra. When you have a mathematical problem involving unknown or changing values, you can use variables to represent them into algebraic expressions, also known as variable expressions. Let's recall what algebraic expressions are.
Algebraic expressions are calculations that contain a combination of numbers, variables and operations symbols.
Algebraic expressions contain at least one variable, and that is what differentiates them from arithmetic expressions. Figure 1 shows an example of an algebraic expression, and its different components.
Fig. 1. Example of an algebraic expression.
Notice that when you see a number next to a variable in an algebraic expression, like \(3x\) in the example above, it represents multiplication. In this case, \(3x\) means \(3\) times the value of \(x\). This is done to avoid confusion between the multiplication symbol \( \times \) and the commonly used \(x\) variable.
Another important concept that you need to understand when working with algebraic expressions, is the concept of terms.
A term can be just a number (constant) or a combination of a number and one or more variables.
An algebraic expression is a combination of such terms, separated by operation symbols.
In the example above,
\[3x + 1\]
\(3x \Rightarrow\) is the first term, where \(3\) is the coefficient, and \(x\) is the variable,
\(1 \Rightarrow\) is the second term, which is a constant.
Remember that, a coefficient is a number that is multiplied by a variable. If a variable doesn't have a coefficient, then it is assumed to be 1.
Evaluating an Algebraic Expression
Having a variable in an expression means that the value of the expression will be different, depending on the value of the variable used to evaluate it.
If you have the algebraic expression \(4x + 5\), and you evaluate it when \(x = 2\). The result will be as follows.
\[4x + 5 = 4 \cdot \textbf2 + 5 = 8 + 5 = 13\]
If you then evaluate it when \(x = 3\), the result will be different, as you can see below.
\[4x + 5 = 4 \cdot \textbf3 + 5 = 12 + 5 = 17\]
Read our article about Linear Expressions to lean more about this topic.
Variables in Algebraic Equations
Algebraic equations differentiate themselves from algebraic expressions, because they contain an equal sign. See an example of an algebraic equation showing all its components in Figure 2 below.
Fig. 2. Example of an algebraic equation.
Notice that in an algebraic equation, you will have a left side and a right side of the equation. The left side of the equation corresponds to the term or combination of terms on the left-hand side of the equal sign, and the right side of the equation corresponds to the term or combination of terms on the right-hand side of the equal sign.
Both sides of the equation need to be equal. Therefore, you solve algebraic equations to find the value of the variable that makes the equation true.
Let's see an example.
Mike ordered a shirt and a pair of shoes. He spent a total of \($100\), and the shirt cost him \($45\). This can be represented with the following mathematical equation:
\[45 + x = 100,\]
where:
\(x \Rightarrow\) is the variable that represents the quantity that we don't know yet, which is the cost of the shoes.
We can solve for \(x\), to find the cost of the shoes. To do this, you need to subtract \(45\) from both sides of the equation.
\[\begin{align}45 + x &= 100 \\ \\\cancel{45} - \cancel{\textbf{45}} + x &= 100 - \textbf{45} \\ \\x &= 55\end{align}\]
The cost of the shoes is \($55\).
Linear Equations with Two Variables
Linear equations can have more than one variable in them. Let's recall what linear equations are.
Linear equations are algebraic equations where the degree of the variables is 1.
The degree of a variable is the superscript number next to the variable. For instance, in \(x^2\) the degree of \(x\) is \(2\). In \(x\), the degree is \(1\), usually omitted from \(x^1\).
Linear equations with two variables can be written in standard form as:
\[ax + by + c = 0,\]
where:
\(a\) and \(b\) \(\Rightarrow\) are real numbers and the coefficients of the variables \(x\) and \(y\),
\(c \Rightarrow\) is a constant.
Here is an example of a linear equation with two variables in standard form:
\[-2x + y -1 = 0,\]
where:
\(a = -2\), \(b = 1\) and \(c = -1\).
You can also find linear equations written in the slope-intercept form as:
\[y = mx + b,\]
where:
\(m\) \(\Rightarrow\) is the slope,
\(b \Rightarrow\) is the \(y\)-intercept.
\(y = 2x + 1\), is a linear equation with two variables written in slope-intercept form, and it graphically represents a straight line.
where:
\(m = 2\), and \(b = 1\).
Read our explanations about Writing Linear Equations, and Solving Linear Equations for more details and examples.
Order of Operations with Variables
The standard order of operations \(PEMDAS\), that you use to solve arithmetic operations, also applies when solving algebraic expressions. Let's recall what the acronym \(PEMDAS\) stands for:
\(P \Rightarrow \) Parentheses
\(E \Rightarrow \) Exponents
\(M \Rightarrow \) Multiplication
\(D \Rightarrow \) Division
\(A \Rightarrow \) Addition
\(S \Rightarrow \) Subtraction
The parentheses are solved first, then the exponents, followed by multiplication and division (done in order from left to right), and finally addition and subtraction, also done in order from left to right.
When solving algebraic expressions, you will need to combine like terms. When two terms have different variables, they are not considered "like terms".
Like terms are terms of the same kind, based on their variables and powers. For example, constants are always like terms with other constants.
Here are a couple of examples to give you some practice.
a) Simplify the algebraic expression \(4x + 2x + 5(2^2 + 1)\), and evaluate it when \(x = 2\).
Following the \(PEMDAS\) rules, you need to solve the operation inside the parentheses first. Inside the parentheses, you have an exponent and an addition, so let's solve them in that order.
\[4x + 2x + 5(4 + 1)\]
\[4x + 2x + 5 \cdot 5\]
Next, you need to solve the multiplication,
\[4x + 2x + 25\]
Now you can combine (add or subtract) like terms. In this case, \(4x\) and \(2x\) are like terms as they have the same variable \(x\).
\[6x + 25\quad \text{This is the simplified algebraic expression}\]
Now we can evaluate it when \(x = 2\),
\[6x + 25 = 6 \cdot 2 + 25 = 12 + 25 = \textbf{37}\]
b) Simplify the algebraic expression \(4x + 2y + 3(x + 2)\), and evaluate it when \(x = 1\) and \(y = 3\).
Again, you need to solve the operation inside the parentheses first. However, \(x\) and \(2\) are not like terms, therefore you cannot add them together. To solve the parentheses in this case, you need to expand them, by multiplying \(3\) by each of the terms inside the parentheses.
\[4x + 2y + 3x + 6\]
Next, you need to combine (add or subtract) like terms,
\[7x + 2y + 6\]
And finally, you can evaluate it for the given values of the variables \(x\) and \(y\).
\[7x + 2y + 6 = 7 \cdot 1 + 2 \cdot 3 + 6 = 7 + 6 + 6 = \textbf{19}\]
Dependent and Independent Variables
Variables in Algebra can be dependent or independent, based on whether their value depends or not on the value of another variable.
Let's look at each case in turn.
Independent variables are variables that do not depend on the value of any other variable.
Here is an example. Think about the following scenario.
If you are paid \($10\) per hour for your job. The salary that you get monthly will depend on how many hours you work in a month.
This can be represented in the following algebraic equation:
\[s = 10h,\]
where:
\(s \Rightarrow \) monthly salary,
\(h \Rightarrow \) number of hours worked in a month.
In this case, \(h\) is the independent variable, because it does not depend on any other variable. You control how many hours you work.
Now let's define what dependent variables are.
Dependent variables are variables whose value depends on the value of other variables.
Going back to our scenario.
In the algebraic equation,
\[s = 10h,\]
The dependent variable is \(s\), because the amount that you receive as monthly salary will depend on the value of the variable \(h\).
For example,
If you work \(80\) hours \((h = 80)\), then your monthly salary will be,
\[s = 10h = 10 \cdot 80 = \textbf{\$800}\]
If you only manage to work \(55\) hours \((h = 55)\), your salary that month will be,
\[s = 10h = 10 \cdot 55 = \textbf{\$550}\]
Variables in Algebra - Key takeaways
- In Algebra, variables are letters that are used to represent unknown values that can change.
- Algebraic expressions are calculations that contain a combination of numbers, variables and operations symbols.
- Algebraic expressions contain at least one variable, and that is what differentiates them from arithmetic expressions.
- Algebraic equations differentiate themselves from algebraic expression, because they contain an equal sign.
- The standard order of operations \(PEMDAS\), that you use to solve arithmetic operations, also applies when solving algebraic expressions.
- Variables in Algebra can be dependent or independent, based on whether their value depends or not on the value of another variable.
References
- Sorell, Tom (2000). Descartes: A Very Short Introduction. New York: Oxford University Press.
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