Experienced teachers know that some students seem to have a natural feel of how to solve an equation. They just know how and when an operation should be applied to both sides of the equation. Capable math students may not be fully aware of why they are using a particular strategy to solve an equation, but they know how to apply the strategy. Students who struggle with solving basic equations ask questions like the following: How do you know whether to add or subtract the same number to both sides of the equation? How do you know whether to multiply or divide both sides of the equation by the same number? How do you know in what order various operations need to be applied to both sides of an equation? What does “find x” or “what is x” really mean? The purpose of this post is to provide answers to these types of questions. Readers who have read the posts Inverses of Relations and Functions and Inverse of a Matrix will immediately see the connection between those posts and this post.
To get started, you can download the free handouts Basic Equation Solving Strategies, Strategies for Solving Exponential and Logarithmic Equations, and Basics of Solving Inequality Relations. These free handouts (and many more) can also be accessed by visiting mathteachersresource.com/instruction-content. Depending on the course I’m teaching, I give one or more of these handouts to my students. Basic Equation Solving Strategies breaks down the basic algebraic equations that students will encounter in a lower level math course into six equation types which I call “case 1” through “case 6”. When solving different types of equations, I routinely ask my students to identify the type of equation being solved. When students can identify the type of equation being solved and know the basic algorithm to solve that type of equation, they can quickly and efficiently solve the equation; no wasted time. In application problems, solving an equation is usually just one small step in finding a solution of a problem.
This post will focus on how to solve a case 1 type of equation by the “work backwards” method. Case 1 equations are equations in which the variable appears only once in the equation. The work backwards technique of solving an equation is well known, but the manner in which it’s presented in this post is different. I use the following analogy to explain the basic reasoning behind the work backwards method: Suppose that you have established a base camp on a camping trip and take a day hike in the woods. You can always get back to your base camp by simplify reversing your steps.
Sample case 1 equations are shown in the text box below. I certainly would not have a beginning algebra student initially solve equations of this complexity; however, it’s my hope that they will eventually learn how to solve equations of this complexity. It takes a while for students to remember that equations involving the squaring or absolute value operations can have two solutions or no solutions. From personal experience, the work backwards method as presented in this post is effective with both low and high ability students. I remember one of my students saying to a classmate, “The work backwards method really works, but you need to know the order of operation rules.” Is the work backwards method effective with all students? Of course not!
The text box below gives five examples of case 1 equation types. When solving for a variable in terms of other variables, think of the other variables as constants.
To use the work backwards method, it’s necessary to understand which operation reverses a given operation. After looking at specific examples of each type of operation, most students quickly develop an understanding of what reversing an operation means. The text box below gives a summary of common operations and the corresponding inverse operation. Note that the reciprocal operation is its own reverse or inverse operation.
There are three major steps in solving a case 1 type of equation.
Step1) Recognize that an equation describes a process that starts with an unknown value of a variable, then performs a series of back-to-back mathematical operations, according to the order of operation rules. This process is described in what I call “the equation solve plan” which may or may not be explicitly stated by the student. Equation solve plans are road maps that lead to the solution of an equation. The student now knows what operation needs to be applied to both sides of the equation and when the operation needs to be applied. Initially, I require beginning students to explicitly state the equation solve plan as shown in the examples below where the solve plan is described in a box to the left of the list of equations. The equation solve plan is the series of back-to-back operations in the equation that are performed, according to the order of operation rules, on the variable being solved for.
Step 2) Follow the equation solve plan in reverse order from the last step to the first step. At each step, apply the reverse or inverse operation to both sides of the equation. Continue working backwards until the solution appears. This process is like peeling back an onion layer by layer.
Step 3) Check numerical solutions by plugging the solutions into the original equation. This is very important because it helps students better understand the equation and catch errors. I have no sympathy for a student who gives an incorrect solution and has not bothered to check the solution. With modern calculators, there is no reason that solutions can’t be checked.
The next six text boxes illustrate how the work backwards method works. The text in the box to the left of the list of equations is the equation solve plan. The operation symbol to the right of a step indicates what operation was applied to both sides of the equation. The check mark certifies that the solution was checked. Notice that the solution of the sixth equation is a somewhat different approach of solving the equation.
The handouts previously mentioned in this post are intended to provide a set of efficient algorithms that students can use to solve the types of equations and inequalities found in lower level math courses. Of course, some students will find a faster way to solve an equation or inequality in a special situation. However, I have observed students who seem to have a knack for making easy problems difficult. One time I saw a student rewrite the equation and then use the quadratic formula to solve the equation. The solution could have been obtained in 3 or 4 steps by using the work backwards method!
Some Personal Observations:
- Equation solving is not an end in itself, but a small step in a larger application problem.
- Practice solving equations is really nothing more than good mental gymnastics.
- Equation solving is an essential skill, but not creative mathematics.
- Discovering an equation that models a law of nature is creative mathematics.
- Discovering a new algorithm to solve a math problem is creative mathematics.
- Students primarily take algebra to start learning to how to reason abstractly with symbols; not how to learn to manipulate polynomial expressions, graph equations and solve equations. Of course, they don’t realize this.
- Professionals such as engineers, scientists, writers, artists, musicians, educators, company managers, business executives, military planners, etc. routinely think and reason abstractly with symbols, but they never or seldom factor a polynomial or use the quadratic formula.
- Beginning students should be required to express solutions in decimal format because an expression like 3 + √(29) has no real meaning for them.
- It’s necessary to remind beginning students that dividing by a number is the same as multiplying by the reciprocal of the number and vice versa.
- When the solution of an equation is an algebraic expression, imagine replacing the single variable that was solved for with the expression. If you carefully study the resulting equation, you will see the equation magically transformed itself into an identity. It’s amazing.