Book cover for Algebra and Trigonometry Real Mathematics, Real People

Algebra and Trigonometry Real Mathematics, Real People

Ron Larson

ISBN #9781305071735

7th Edition

6,909 Questions

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234,597 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

Section 2.1 covers the process of solving linear equations including those with fractional expressions while emphasizing the importance of checking for extraneous solutions. It illustrates the method of translating real-world scenarios into mathematical models and applying common formulas to tackle problems in geometry, finance, and science. Through step-by-step examples, students learn how careful manipulation of equations, along with diagramming and ratio analysis, is critical for obtaining accurate results and verifying solution validity.

Learning Objectives

1

Solve linear equations that include fractional expressions and identify extraneous solutions.

2

Translate real-life scenarios into verbal models and then into mathematical (algebraic) equations.

3

Apply common formulas for area, perimeter, volume, temperature conversion, and interest to solve practical problems.

4

Develop problem-solving strategies that include checking solutions and using graphical methods for validation.

Key Concepts

CONCEPT

DEFINITION

Linear Equation

An algebraic equation in which the highest power of the variable is one, typically written in the form ax + b = 0.

Fractional Expression

An expression that involves fractions in which the numerator, the denominator, or both contain algebraic expressions.

Extraneous Solution

A solution derived from the process of solving an equation that does not satisfy the original equation, often introduced when multiplying or dividing by a variable expression.

Mathematical Modeling

The process of forming a verbal description into a verbal model and finally translating it into an algebraic equation that represents a real-life situation.

Common Formulas

Pre-established equations used in a variety of contexts (e.g., geometry, finance, physics) that can be rearranged to solve for unknown quantities.

Example Problems

Example 1

Fill in the blank. A (n) ________ is a statement that two algebraic expressions are equal.

Example 2

Fill in the blank. A linear equation in one variable is an equation that can be written in the standard form _______.

Example 3

Fill in the blank. When solving an equation, it is possible to introduce a(n) ______ solution, which is a value that does not satisfy the original equation.

Example 4

Fill in the blank. Many real-life problems can be solved using ready-made equations called _______.

Example 5

Fill in the blank. Is the equation $x+1=3$ an identity, a conditional equation, or a contradiction?
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Step-by-Step Explanations

QUESTION

How do you verify that a solution obtained from an equation with fractional expressions is valid?

STEP-BY-STEP ANSWER:

Step 1: Write down the original equation.
Step 2: Substitute the found value for the variable into every term of the equation.
Step 3: Carefully evaluate each fractional expression making sure that denominators are not zero.
Step 4: Simplify both sides of the equation and compare the results.
Step 5: If both sides are equal, the solution is valid; otherwise, it is extraneous.
Final Answer: The solution is confirmed if substitution results in a true statement, otherwise it is rejected as extraneous.

Checking a Solution Involving Fractions

QUESTION

How can you use the properties of similar triangles to find the height of a building given the building's shadow and the shadow cast by a known object?

STEP-BY-STEP ANSWER:

Step 1: Draw a diagram of the situation with the building and a post, labeling the heights and respective shadow lengths.
Step 2: Set up a ratio using the property that corresponding sides of similar triangles are equal. For example: (Height of building)/(Length of building's shadow) = (Height of post)/(Length of post's shadow).
Step 3: Substitute the known values into the ratio.
Step 4: Solve for the unknown (height of the building) by cross-multiplying.
Step 5: Check that the computed height is reasonable using unit analysis and verification.
Final Answer: The calculated height from the ratio is the height of the building if the ratio holds true.

Using Similar Triangles to Determine Height

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Common Mistakes

  • Failing to check for extraneous solutions after multiplying or dividing by variable expressions.
  • Not converting percentage values into decimals when using formulas for profit or interest.
  • Incorrectly setting up ratios when using similar triangles for measurement problems.
  • Overlooking restrictions on the variable, such as values that cause a denominator to become zero.
  • Mistakenly assuming that a solution validated graphically is free of computational errors.