Algebraic expressions and equations will be used when we wish to describe and demonstrate geometric facts and formulas involving lengths, areas, and other numerical quantities. The use of variables generalizes the statement so that it describes all relevant situations, not just a single example. Hence, it is essential that we can simplify expressions and solve equations.
An expression in mathematics is a representation of a numerical quantity consisting of a finite number of variables and/or numbers joined by algebraic operations such as addition, subtraction, multiplication, division, and exponentiation. An expression is said to be algebraic if it contains at least one variable. We will also use the term numerical expression to describe any representation that is either an algebraic expression or number.
An equation is a statement formed by joining two algebraic expressions together with an equal sign. When we solve an equation, we determine which numerical quantities will make the statement true.
The equals sign is also used when we simplify a statement, joining the equivalent expressions immediately preceding and following the sign. In this situation, a series of equal signs in succession are allowed. You show verify that any expressions joined by this symbol are indeed equal.
because the two expressions on either side of the first equal sign are not equal, the expression \(2xy\) has been subtracted in the second expression, but not the first.
When solving an equation, it is bad practice to put equal signs between steps as the quantity frequently changes. Each step of the solving process should be a separate statement with a single equal sign. Simply write each step on a new line or use arrows or words (then, so, since, etc.) to link the statements.
We have found two numbers \(5\) and \(-5\) that solve the equation. If this were a problem about geometric quantities, we frequently rule out negative numbers as unreasonable.
As we share our solutions with others, it is important to use the equals sign appropriately. Check that the two expressions on either side of the symbol \(=\) are really equal to each other.
A frequent error made by students is to link steps of a computation with equals signs when the two expressions are not equal to each other. Typically, an operation has been performed in a step which did not exist previously. For example, a student simplifying \((x+3)^2-6x\) may write
Why is this incorrect? Near the end of the line, we see \(25=x=\pm 5\text{.}\) This would imply that \(25=\pm 5\text{,}\) a statement that is clearly false. It appears that the student has taken the square root of both sides of \(x^2=25\) but that operation changes the value. While \(x=\pm 5\) is the solution of \(x^2=25\text{,}\) the expression \(x\) is not equal to \(25\) so they should not be linked with an equals sign.
Rather than linking steps with equal signs, you may write them as a list of equations as shown in the answer to Example A.1.15 or use words or arrows between steps.
