Subsection A.1 Algebraic Expressions and Equations
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.
Definition A.1.7.
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.
Note the difference between an expression and an equation.
Definition A.1.8.
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.
Example A.1.9. Simplifying an Expression.
Simplify the expression \((x+y)^2-2xy\text{.}\)
Answer.
Solving this expression requires an expansion of \((x+y)^2\) before we can subtract \(2xy\text{.}\) One correct way of writing this is the following.
\begin{equation*}
(x+y)^2-2xy=(x+y)(x+y)-2xy=x^2+2xy+y^2-2xy=x^2+y^2.
\end{equation*}
One might also write \((x+y)^2=(x+y)(x+y)=x^2+2xy+y^2\) and then compute \((x+y)^2-2xy=x^2+2xy+y^2-2xy=x^2+y^2.\)
\begin{equation*}
(x+y)^2=x^2+2xy+y^2-2xy=x^2+y^2
\end{equation*}
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.
Example A.1.10. Solving an Equation.
Solve the equation \((x+3)^2-6x=34.\)
Answer.
One acceptable way of writing this for exercises is the following:
\begin{align*}
(x+3)^2-6x\amp =\amp 34\\
x^2+6x+9-6x\amp =\amp 34\\
x^2+9\amp =\amp 34\\
x^2\amp=\amp 25\\
x \amp = \pm 5
\end{align*}
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.
Checkpoint A.1.12. Find the Error.
The example in
Remark A.1.11 also suggests that
\(34=25\text{.}\) Identify where the value changes and why the equal sign is misleading.
Answer.
After the equation \(x^2+9=34\text{,}\) the number \(9\) is subtracted changing the value.
Rather than linking steps with equal signs, you may write them as a list of equations as shown in the answer to
Example A.1.10 or use words or arrows between steps.
