Book cover for Prealgebra

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

ISBN #9781938168994

1st Edition

5,729 Questions

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210,912 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section emphasizes solving linear equations by maintaining balance, using properties of equality such as addition, subtraction, multiplication, and division. It involves simplifying expressions, translating word problems into equations, and verifying the solutions by substitution. These foundational skills are essential for both academic problem-solving and practical applications in various careers.

Learning Objectives

1

Solve linear equations using the Subtraction, Addition, Multiplication, and Division Properties of Equality.

2

Simplify equations by combining like terms and using the distributive property before isolating the variable.

3

Translate word problems into algebraic equations and solve them.

4

Verify solutions by substituting values back into the original equations.

Key Concepts

CONCEPT

DEFINITION

Equation

A mathematical statement asserting that two expressions are equal.

Solution

A value for the variable that makes the equation a true statement.

Subtraction Property of Equality

If a = b, then a - c = b - c for any real number c.

Addition Property of Equality

If a = b, then a + c = b + c for any real number c.

Multiplication Property of Equality

If a = b, then ac = bc for any real number c.

Division Property of Equality

If a = b and c ≠ 0, then a/c = b/c.

Example Problems

Example 1

In the following exercises, determine whether the given value is a solution to the equation. Is $\quad y=\frac{1}{3} \quad$ a solution of $4 y+2=10 y ?$

Example 2

In the following exercises, determine whether the given value is a solution to the equation. Is $\quad x=\frac{3}{4} \quad$ a solution of $5 x+3=9 x ?$

Example 3

In the following exercises, determine whether the given value is a solution to the equation. Is $\quad u=-\frac{1}{2} \quad$ a solution of $8 u-1=6 u ?$

Example 4

In the following exercises, determine whether the given value is a solution to the equation. Is $\quad v=-\frac{1}{3} \quad$ a solution of $9 v-2=3 v ?$

Example 5

In the following exercises, solve each equation. $$ x+7=12 $$
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Step-by-Step Explanations

QUESTION

How do you solve the equation x + 11 = -3 using the Subtraction Property of Equality?

STEP-BY-STEP ANSWER:

Step 1: Identify the operation affecting the variable. Here, 11 is added to x.
Step 2: Use the Subtraction Property of Equality to subtract 11 from both sides.
Step 3: Write the new equation: x + 11 - 11 = -3 - 11.
Step 4: Simplify both sides to find x = -14.
Final Answer: x = -14.

Solving x + 11 = -3

QUESTION

How do you solve the equation m - 4 = -5 using the Addition Property of Equality?

STEP-BY-STEP ANSWER:

Step 1: Identify that 4 is subtracted from m.
Step 2: Add 4 to both sides of the equation to undo the subtraction.
Step 3: Write the new equation: m - 4 + 4 = -5 + 4.
Step 4: Simplify to obtain m = -1.
Final Answer: m = -1.

Solving m - 4 = -5

QUESTION

How do you translate and solve the sentence 'five more than x is equal to 26' into an equation?

STEP-BY-STEP ANSWER:

Step 1: Translate the wording into an algebraic equation: x + 5 = 26.
Step 2: Use the Subtraction Property of Equality by subtracting 5 from both sides.
Step 3: Write the new equation: x = 26 - 5.
Step 4: Simplify to obtain x = 21.
Final Answer: x = 21.

Translating 'five more than x is equal to 26'

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Common Mistakes

  • Failing to perform the same operation on both sides of the equation, which breaks the balance.
  • Miscalculating when combining like terms or simplifying fractions and decimals.
  • Incorrectly translating word problems into algebraic equations.
  • Neglecting to check solutions by substitution to ensure they satisfy the original equation.