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# Solving Equations

## Look at different strategies for solving equations. Topics include the different meanings attributed to the equal sign and the strengths and limitations of different models for solving equations. Explore the connection between equality and balance, and practice solving equations by balancing, working backwards, and inverting operations.

### In This Session

Part A: Equality and Balance
Part B: False Position and Backtracking
Part C: Bags, Blocks, and Balance
Homework

In the previous session, we looked at linearity in different situations. We used spreadsheets to work with linear functions in tables, equations, and graphs. We also explored connections between rates and slopes, and examined the role of independent and dependent variables in linear functions. Now that we have explored characteristics of equations for lines, we will develop strategies for solving linear equations in this session.
Note 1

### Learning Objectives

In this session we will explore different methods for solving equations. We will:

• Explore the connection between equality and balance
• Solve equations by balancing, working backwards, and inverting operations
• Explore the strengths and limitations of different models for solving equations
• Solve systems of equations

### Key Terms

Previously Introduced:

Axes: The axes of a graph are the base (or zero) values of two quantities that are being compared in a coordinate graph. The horizontal axis is often referred to as the x-axis and the vertical axis is often referred to as the y-axis.

Function: A function is any relationship between inputs and outputs in which each input leads to exactly one output. It is possible for a function to have more than one input that yields the same output.

Variable: The term variable can have different meanings:

• An indeterminate, a symbol used to represent generalized arithmetic (such as 0 x a = 0)
• An unknown, used to represent a particular number (such as a + 3 = 7)
• A relationship between quantities (such as a + b = 10)
• A term used to help in understanding the study of algebraic structures (such as a + b = b + a)

Slope: The slope of a line is often described as the ratio of rise to run. Slope is also the amount that the dependent variable changes for each increase by one in the independent variable. The formula for slope is: slope = (change in y) / (change in x).

Whole Number: A whole number is a counting number or zero. The whole numbers can be written as the series 0, 1, 2, 3, … . Negative numbers and fractions are not included in the set of whole numbers.

New in This Session:

Backtracking: Backtracking is a method of solving equations that involves undoing the operations in an equation to work backward from an output to an input.

Covering Up: Covering up is a method of solving equations by covering up a portion of the equation to yield a simpler, one-step equation. For example, if 6 + 2m = 12, cover up 2m to give the equation 6 + ? = 12. Because 6 + 6 = 12, the ? must be 6, so 2m = 6 and m = 3.

Equivalence: A relationship between two quantities, usually represented by the equals sign “(=)”. There are three important properties of equivalence:

• Every quantity is equivalent to itself (3 = 3)
• Equivalency does not depend on the sides of the equation (if 2 + 3 = 5, then 5 = 2 + 3)
• With three quantities a, b, and c, if a = b and b = c, then a = c (if 6 + 9 = 15 and 15 = 5 x 3, then 6 + 9 = 5 x 3)

False Position: False postion is a method of solving equations by assuming a convenient number is the solution, then adjusting that number to find the actual solution.

Intercept: An intercept is an intersection of a graph with one of the axes. An intersection with the horizontal axis is often referred to as an x-intercept, and an intersection with the vertical axis is often referred to as a y-intercept.

System of Equations: A system of equations is a set of two or more equations with a common solution. Systems of equations may have multiple variables and multiple solutions. If each equation of a system is graphed, the solutions will be any points where all the graphs intersect.

Integer: An integer is any number which is a counting number, zero, or the opposite of a counting number. The set of integers is sometimes written as {…, -3, -2, -1, 0, 1, 2, 3, …}.

Set: A set is a list of items, or elements, placed in brackets, for example, {2, 3, 4} or {red, blue, yellow}. Sets may contain an unlimited number of elements: {0, 1, 2, 3, …}. The set containing no elements, {}, is called the empty set or null set.”

Solution Set: The solution set to an equation is the set of all values for the variables in the equation that make the equation true. Here are some examples:

• The equation x + 5 = 9 has the solution set {4}, since x = 4 is the only way to make the equation true.
• The equation x2 = 9 has the solution set {3, -3}. This represents the two different numbers that x can be.
• The equation x + 3 = x + 4 has the solution set {}. This means that there are no values for x which make the equation true. The set {} is referred to as the null set, or empty set.
• The equation 2x + y = 10 has an infinite number of solutions, including (5, 0), (4, 2), and many other ordered pairs of values. The solution set would be written as {(5, 0), (4, 2), …}.

### Notes

Note 1

The problems in this session progress from informal to formal strategies for solving equations. One of the initial stumbling blocks to solving equations is making sense of the equal sign. In Part A, we will explore different interpretations of the equal sign, differentiating between the result of a process and an equivalence relation.

Later in the session, we will introduce historical and informal strategies for solving equations, such as the method of false position. In this method, we’ll take a guess and then modify that guess so that it is a solution to the equation.

We’ll also introduce the method of backtracking. This method involves solving equations by reasoning backwards from the answer, undoing the operations in reverse order. This is an informal method that seems intuitive to many students. It coincides with a view of equations primarily as a process. (Take a number, multiply it by 2, and add 1. The result is 12. What is the number?)

The traditional method of solving a linear equation by doing the same thing to both sides belies a more static interpretation of an equation. Picturing this method as a series of bags and blocks on a scale can help us to think of algebraic expressions as objects. As the equations get more complicated, however, the balance model becomes less appropriate, just as backtracking does not work with certain equations.

Review

Groups: Discuss any questions that came up on the homework. Be sure to look at solutions to the “undoing” problem (Problem H4, Session 5), because “undoing,” or solving, equations will be the focus of today’s session.