Evaluating Expressions
Evaluating expressions is a fundamental concept in mathematics involving the substitution of values into mathematical expressions to determine their value. An expression typically consists of variables, constants, and operations, with variables representing unknown quantities. The evaluation process is essential for solving equations, particularly in algebra, where one or more variables need to be defined.
The history of algebra illustrates its evolution from rhetorical forms—where equations were expressed in full sentences—to modern symbolic notation, which simplifies complex relationships. Evaluating expressions can be done with one variable, where a specific value is substituted and the operations are performed according to the established order of operations (parentheses, exponents, multiplication and division, addition and subtraction). For expressions with two variables, the evaluation process is similar but requires substituting values for each variable sequentially.
Additionally, equivalent expressions yield the same result when evaluated, and function notation provides a contemporary way to express relationships between variables. Understanding how to evaluate expressions is crucial for problem-solving in various mathematical contexts, enabling clearer insights and solutions.
Evaluating Expressions
An expression is a mathematical sentence made up of variables and terms, such as 3x and 2y, as well as constants, such as 3 and 7. Terms are often separated by operations, such as + and −, and each variable stands for a number that begins as unknown. In 3x + 6, for example, x is the variable, and the more complicated 2x ( 3 + y ) has both x and y as variables.
Overview
Throughout the history of algebra, symbolism went through three different stages. Rhetorical algebra, in which equations were written as full sentences ("One thing plus two things equals three things"). This was how the ancient Babylonians expressed their mathematical ideas, and it stayed this way until the sixteenth century. Then came syncopated algebra, in which the symbolism became more complex but still didn’t contain all the qualities of today’s symbolic algebra. Restrictions, such as using subtraction only on one side of an equation was lifted when symbolic algebra came along and Diophantus' book Arithmetica was eventually replaced by Brahmagupta's Brahma Sphuta Siddhanta. Eventually, Ibn al-Banna, François Viète and René Descartes completed the modern notation used today, which is known as Cartesian geometry.
Most expressions have at least one unknown variable that is a symbol or letter, frequently x or y. These variables are typically used as a way of solving a problem and determining the value of something that isn’t known yet. Unknown variables can be defined according to what they are added to, subtracted from, multiplied by, or divided by. An example of an unknown variable that has been added to something is x + 89 = 137 where 89 had been added to x and it is known that the sum equals 137. The variable x is unknown, but it can be found by performing the subtraction of 89 from 137. Similarly in multiplication an unknown variable can be multiplied by 5 so that 5y = 75. The variable y is still unknown, but 75 can be divided by 5 to find the value of y is 15. The process of solving for the unknown variable is called evaluating the expression, and it can be done with one or two variables depending on the complexity of the expression.
Evaluating Expressions with One Variable
When evaluating an expression with one variable, a number is simply plugged into the only variable in the expression, like turning x into 3, or replacing x with 3, as in the evaluation of 3x + 6. When x = 3 the expression becomes 3(3) + 6, which is also means 9 + 6, which is also the same as 15. This means that the expression 3x + 6 has been evaluated as being 15 when x = 3. Whenever a specific value is substituted in for a single variable, and the operations are performed, this is called evaluating an expression with one variable.
The steps for evaluating an expression always begin with substituting each letter with an assigned value. Then the numbers are enclosed with parenthesis when they are being multiplied to other numbers, and the order of operations is used to determine which step must take place first, second, and third. Understanding the order of operations is very important. If there aren’t any exponents, then multiplication and division are done next. Finally, addition and subtraction are done last in the order of operations and the expression is evaluated. In mathematics, the order of operations is the basic rule that determines which procedures are done first: exponents and roots, multiplication and division; addition and subtraction.
Often, it is easier to treat division as multiplication by creating the reciprocal or inverse of a number that is being used in division. For example, it may be easier to view 2 ÷ 4 as being the same as 2 × 1/4. The same is often done with subtraction being changed into addition so that 2 – 4 can be viewed as "2 + (−4). Whether these tricks are used or not, the order of operations always remains the same, and the evaluation proceeds according to the three steps above.
An example using exponents might look like 5x2 – 12x + 12 where x = 4. First, the variable is substituted with 4 to get "5(4)2 − 12(4) + 12." Then the order of operations dictates that exponents come before multiplication or addition. Squaring the 4 then gives 16 so the expression becomes 5(16) – 12(4) + 12. Next in the order of operations is multiplication and division, so the expression becomes (5 × 16) – (12 × 4) + 12, which means 80 – 48 + 12. The final order of operations can then be done with addition and subtraction to get 44. This means that the evaluation results in 5x2 − 12x + 12 = 44 when x = 4.
Evaluating Expressions with 2 Variables
Evaluating expressions with two variables is exactly the same as evaluating only one variable but the process is done twice. An example is the evaluation of 3x + 4y when x = 2 and y = 6. First, there is a replacement of x with 2 so that the result is 3(2) + 4y. Then y is replaced with 6 to get 3(2) + 4(6). There are no exponents so the first order of operations is multiplication, giving 6 + 24 so that the final order of operation can be done in addition to get 30. Therefore 3x + 4y = 30 when x = 2 and y = 6.
Equivalent Forms of Expressions
Expressions are equivalent if their evaluations are equal. When two expressions are placed on each side of the = sign, and the same number results for every substitution and of all variables, then an equivalent form of expression is the result. For example, x + 3 and y – 7. If the two expressions are placed on each side of an = sign, the two expressions are said to be equivalent. Equivalent forms of expression can be extremely complex, as in Figure 1. No matter how complex these expressions get, they are still referred to as equivalent if they stand on both sides of an = sign. This form of expression is also known as an equation.
Understanding Function Notation
Function notation is another way of expressing equations as functions. Instead of the previous form already discussed as y = x + 2, a new notation f(x), which is pronounced as "eff-of-eks," was introduced by Leonhard Euler in 1734. The parentheses in this notation do not indicate multiplication as might first be suspected. Instead, the f(x) symbol is really just a substitution for y and written in a different way. It expresses the variable y in terms of x and is therefore written as f(x). With this new notation, it can be shown how substitutions are being made in the expressions and, where the previous variable y didn’t suggest anything about its partner on the other side of the equation, it now suggests a relationship to x that can be visualized from both sides.
For example, when x = 3 it may also be said that y = 3 + 2, but this makes x vanish completely from the equation. Now, with function notation, the equation is written as f(x) = x + 2 and the substitution becomes f(3) = 3 + 5. Function notation shows another way of writing out the same expressions by showing a relationship that was previously hidden from view. It is simple and easy, and there is no need to complicate the new notation beyond its simple purpose.
Evaluating Expressions with Function Notation
When evaluating functional notation, the term f(x) is first shown instead of the old term y but, just as an expression would be evaluated in y, it is also evaluated in f(x). For example, f(x) = x2 + 2x – 1. This entire statement is actually an equation, but the limited expression x2 + 2x −1 can be evaluated by substituting in a value for x once f(x) is given a more clear value. For example, it might be suggested that an evaluation be done for f(2) instead of the previous manner of saying that y = 2. In this case, f(2) = (2)2 + 2(2) – 1, which is the same as saying that f(2) = 4 + 4 – 1 and that f(x) = 7.
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