Algebra in Words: A Guide of Hints, Strategies and Simple Explanations (2014)
SOLVING SIMPLE ALGEBRAIC EQUATIONS
Solving a Simple Algebraic Equation with One Variable (First Degree)
The goal is to completely isolate the variable and to have it equal a number, which is the answer. Although this may seem easy (and it will become easier as you practice), it can also be complicated (if not just tedious), and students who are learning this for the first time often underestimate the importance of doing this in an orderly, systematic way. If you don’t learn to do this properly, you will quickly get left behind in class.
This is done in two general steps:
1. Isolating the “term with the variable,”
o Using addition and subtraction, and then
2. Isolating the variable
o Using division or multiplication.
The following is a chronological list of detailed instructions to help you.
1. If there are any denominators, find the least common denominator and multiply all terms on both sides by the LCD to eliminate all denominators.
2. Simplify: Identify and combine like-terms, if any.
Note: #s 1 & 2 are interchangeable. You can eliminate denominators first and combine like-terms next. It is usually easier to “get rid” of fractions first, so you don’t have to go through Adding & Subtracting Fractions.
3. Isolate the “term with the variable.” Use addition or subtraction to “move*” the constants (non-variable numbers) to the right of the equal sign and…
· Use addition or subtraction to “move*” the term(s) containing variables to the left of the equal sign.
* I use “move” in a context which indicates the use of the addition principle of equality in which you add the opposite of the term you want to move (because adding opposites equal zero, canceling out a term), and what you add/subtract to one side of the equal sign, you must do to the other side to maintain the equality.
4. Simplify by combining like-terms. There should be one term (the term with the variable) on one side and a number on the other side.
5. Isolate the variable: Multiply both sides by the reciprocal of the coefficient in front of the variable. If the coefficient in front of the variable is negative, you should multiply both sides by the negative reciprocal in order to eliminate the negative sign and make your isolated variable positive.
You should now be left with a variable equaling a number.
Arrangement: Descending Order
It is always best to put all terms in descending order – from highest power to lowest power, from left to right. This organization facilitates easier simplification. Put all terms in descending order, even terms within parentheses and groups.
One reason (descending) order matters is for factoring. It is easiest to factor polynomials (like trinomials into binomials) when you see the terms in descending order. It will also help you identify and cancel out common factors (when they are polynomials) when the factors inside the parentheses are in descending order.
Another reason terms need to be in descending order is for long division of polynomials. Because of the systematic process of long division, the divisor and dividend must both be in descending order.
Expressions vs. Equations
It is very important to know the subtle difference between an expression and an equation. Simply put, expressions are not equations. Expressions are combinations of terms and operation symbols with no equal signs. Since expressions do not have equal signs, they cannot be solved, they can only be simplified. Equations are solved. Books often focus on expressions to stress and practice simplification. This is necessary (although sometimes misleading… I’ll explain why shortly), because equations contain expressions.
Equations are mathematical sentences that contain an expression or expressions, and equals signs, and can be solved. The steps to solving an equation involve simplification of the expressions within the equation.
As I was saying above, books focus a great deal on expressions, which is fine, but this is why it can be misleading. The books go by a bottom-up approach and narrow focus on simplifying (factoring) expressions only, before incorporating those applications towards solving equations. What happens is: students get in the mindset of simplifying or factoring an expression (only), and stopping, that (later during solving) after simplifying, they forget to solve the rest, usually not more than two simple steps from the end-point. This is especially evident when students are supposed to solve quadratic equations. A common mistake is that students will successfully factor the trinomial in an equation but then forget to solve for the variables. So keep this goal in mind:
Factoring an expression is only part of solving an equation. Once you successfully factor an expression, get in the habit of continuing on to solve the equation. Those steps are discussed in the next section.
Author’s Note: If it were up to me (and someday, I hope it is… I hope to write an entire algebra text book), the lessons on learning factoring and solving would be consolidated into one lesson, to better connect the reasons for learning factoring to solving equations and graphing. In the meantime, I hope this book helps you realize that these concepts are closely connected and not just separate entities.
Below are simple examples of an expression and an equation. Notice the small details which set them apart.
An expression: 3x2 + x - 10
An equation: 3x2 + x – 10 = 0, or
Also an equation: y = 3x2 + x - 10