Algebraic Expressions
Now that we’ve got the hang of using real numbers and the order of operations, let’s step up our game! So far, the mathematical expressions we have seen have involved real numbers only.
In mathematics, we may see expressions such as [latex]x+5,\frac{4}{3}\pi {r}^{3}[/latex], or [latex]\sqrt{2{m}^{3}{n}^{2}}[/latex]. In the expression [latex]x+5, 5[/latex] is called a constant because it does not vary and x is called a variable because it does. An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division. For example, [latex]3x + 2y - 7[/latex] is an algebraic expression that contains two variables [latex]x[/latex] and [latex]y[/latex] and three constants [latex]3[/latex], [latex]2[/latex], and [latex]7[/latex].
constant, variable, algebraic expression
- A constant is a fixed value or a number that does not change in a particular context.
- A variable is a symbol that represents a value or quantity that can change or vary in a given situation or context.
- An algebraic expression is a mathematical phrase or combination of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division.
Evaluating Algebraic Expressions
Evaluation of an algebraic expression involves a specific process where we replace each variable in the expression with a given number, and then calculate the result using the established order of operations.
- Replace each variable in the expression with the given value
- Simplify the resulting expression using the order of operations
Note: If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.
- [latex]x+5[/latex] for [latex]x=-5[/latex]
- [latex]\frac{t}{2t - 1}[/latex] for [latex]t=10[/latex]
- [latex]\dfrac{4}{3}\pi {r}^{3}[/latex] for [latex]r=5[/latex]
- [latex]a+ab+b[/latex] for [latex]a=11,b=-8[/latex]
- [latex]\sqrt{2{m}^{3}{n}^{2}}[/latex] for [latex]m=2,n=3[/latex]
Algebraic Formulas
An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation [latex]2x+1=7[/latex] has the unique solution [latex]x=3[/latex] because when we substitute [latex]3[/latex] for [latex]x[/latex] in the equation, we obtain the true statement [latex]2\left(3\right)+1=7[/latex].
A formula is an equation expressing a relationship between constant and variable quantities. Very often the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area [latex]A[/latex] of a circle in terms of the radius [latex]r[/latex] of the circle: [latex]A=\pi {r}^{2}[/latex]. For any value of [latex]r[/latex], the area [latex]A[/latex] can be found by evaluating the expression [latex]\pi {r}^{2}[/latex].
Equations and formulas
- An equation is a mathematical statement that shows the equality of two expressions, typically separated by an equal sign. It states that the two expressions have the same value, and the values of variables that make the equation true are called solutions.
- A formula is a mathematical expression that represents a relationship or a rule between variables or quantities. It usually contains variables, constants, and arithmetic operations, and is used to calculate or derive a particular result or value.
