Lesson 7.5: Algebra Strategy: Substitution and Picking Numbers

Introduction

In this lesson, we will explore two powerful strategies for solving algebra problems on the GRE: substitution and picking numbers. These techniques can simplify complex problems and help you identify the correct answer quickly. By the end of this lesson, you will be able to apply these strategies effectively to different types of algebra problems.

Learning Objectives

• Plugging in answer choices to backsolve problems.
• Picking smart numbers for variable expressions.
• Knowing when these tactics beat direct algebra.
• Backsolving multiple-choice algebra problems efficiently.
• Choosing convenient numbers to test relationships.

Substitution Strategy

Substitution is an effective method for solving algebra problems, especially in multiple-choice settings. This approach involves replacing variables with specific values (often from answer choices) to see which one satisfies the problem conditions.

When to Use Substitution

Substitution is particularly useful in the following scenarios:

1. When the problem contains multiple answer choices.
2. When you are dealing with complex equations where direct solving is cumbersome.
3. When functions or expressions involve variables that can be easily managed by plugging in values.

Example 1: Backsolving a Multiple-Choice Problem

Consider the following problem:

_A bag contains some red and blue marbles. If there are a total of 15 marbles in the bag and the number of red marbles is twice the number of blue marbles, how many red marbles are there?_

Choices: 5, 6, 8, 10

Solution:

Let $ r $ represent the number of red marbles and $ b $ the number of blue marbles. The relationships can be expressed as follows:

1. $ r + b = 15 $
2. $ r = 2b $

Instead of solving these equations directly, we can use substitution with the answer choices. We can compute the values based on each answer choice to see which satisfies the conditions of the problem.

1. If $ r = 5 $:

Then $ b = 15 - 5 = 10 $ and $ r = 2b \Rightarrow 2 \cdot 10 = 20 $ (not possible)

1. If $ r = 6 $:

Then $ b = 15 - 6 = 9 $ and $ r = 2b \Rightarrow 2 \cdot 9 = 18 $ (not possible)

1. If $ r = 8 $:

Then $ b = 15 - 8 = 7 $ and $ r = 2b \Rightarrow 2 \cdot 7 = 14 $ (not possible)

1. If $ r = 10 $:

Then $ b = 15 - 10 = 5 $ and $ r = 2b \Rightarrow 2 \cdot 5 = 10 $ (possible)

Thus, the correct answer is $ 10 $ red marbles. This problem illustrates how substitution can quickly narrow down the choices and confirm the correct answer.

Picking Numbers Strategy

The picking numbers strategy is another powerful tool for solving algebra problems. This method involves selecting specific numbers to represent variables that simplify the situation and help you test relationships or evaluate expressions.

When to Use Picking Numbers

This technique is particularly effective when:

1. The problem involves variables that can take on various values.
2. You need to evaluate expressions where simplification might clarify the relationships.
3. The exact values of the variables are less important than the relationships between them.

Example 2: Evaluating an Expression

Suppose you are asked to determine the value of the expression $ x^2 - 3x + 2 $ when $ x = 4 $.

Solution:

Instead of plugging in directly, let’s understand how this expression behaves. Breaking down the factors:

$$egin{align}\text{The expression is: } & x^2 - 3x + 2 \ \ \text{Factorization results in: } & (x - 1)(x - 2) \end{align}$$

Now let's test a few numbers. We can evaluate the expression at different $ x $ values, including lower values like $ 1 $, $ 2 $, $ 3 $ in addition to $ 4 $ to understand the behavior of the expression in relation to its roots.

1. For $ x = 1 $:

$ 1^2 - 3(1) + 2 = 0$

(The expression equals zero)

1. For $ x = 2 $:

$ 2^2 - 3(2) + 2 = 0$

(The expression equals zero)

1. For $ x = 3 $:

$ 3^2 - 3(3) + 2 = 2$

(The expression yields a positive results)

1. For $ x = 4 $:

$ 4^2 - 3(4) + 2 = 2$

(The expression yields a positive result)

By observing the behavior of the polynomial, you can quickly evaluate how it transitions around its roots without needing the exact calculations at every step. Picking numbers helps explore the relations between variables and their expressions.

Conclusion

Substitution and picking numbers are two essential strategies for tackling GRE algebra problems. By understanding how and when to apply these methods, you can approach a variety of algebraic challenges more efficiently. Always consider substitution when faced with answer choices and picking numbers when simplifying expressions with variables.

Study Notes

• Substitution is useful for backsolving multiple-choice problems.
• Test each answer choice by substituting to quickly find the correct one.
• Use picking numbers to evaluate expressions without fully solving the equations.
• Choose simple, convenient numbers for variables to simplify complex relationships.
• Confirm your findings by checking how specific values affect given equations or expressions.