Book cover for Elementary Algebra

Elementary Algebra

Lynn Marecek

ISBN #9780998625713

1st Edition

6,645 Questions

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32,192 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section emphasizes the balance required in an equation, paralleling the balance of physical objects like rocks. By applying the addition, subtraction, division, and multiplication properties of equality, one can systematically isolate variables and solve linear equations. Additionally, the ability to translate verbal descriptions into algebraic equations is crucial for solving real-world problems. A key takeaway is always checking your solution by substituting it back into the original equation to ensure the balance (equality) remains intact.

Learning Objectives

1

Verify and check solutions to linear equations to ensure correctness.

2

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

3

Simplify and solve equations involving variables and constants on both sides, including those with fractions or decimals.

4

Translate real-world and word problems into algebraic equations and solve for the unknown.

5

Apply the Division and Multiplication Properties of Equality to isolate variables in 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 true when substituted back into it.

Subtraction Property of Equality

If a = b, then a − c = b − c; subtracting the same amount from both sides maintains equality.

Addition Property of Equality

If a = b, then a + c = b + c; adding the same amount to both sides maintains equality.

Multiplication Property of Equality

If a = b, then a · c = b · c, provided c is not zero; multiplying both sides by the same nonzero number maintains equality.

Division Property of Equality

If a = b, then a ÷ c = b ÷ c, provided c is not zero; dividing both sides by the same nonzero number maintains equality.

Simplification

The process of combining like terms and performing arithmetic operations to reduce an expression to its simplest form.

Translation

The process of converting a word problem or everyday scenario into an algebraic equation.

Example Problems

Example 1

$$ \begin{aligned} &\underline{1} . \text { Is } y=\frac{5}{3} \text { a solution of }\\ &6 y+10=12 y ? \end{aligned} $$

Example 2

Is $x=\frac{9}{4}$ a solution of $4 x+9=8 x ?$

Example 3

$$ \begin{aligned} &\underline{3} . \text { Is } u=-\frac{1}{2} \text { a solution of }\\ &8 u-1=6 u ? \end{aligned} $$

Example 4

Is $v=-\frac{1}{3}$ a solution of $9 v-2=3 v ?$

Example 5

In the following exercises, solve each equation using the Subtraction and Addition Properties of Equality. $$ x+24=35 $$
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Step-by-Step Explanations

QUESTION

Solve the equation: y + 37 = -13

STEP-BY-STEP ANSWER:

Step 1: Identify the operation to isolate y. The equation adds 37 to y.
Step 2: Use the Subtraction Property of Equality by subtracting 37 from both sides: y + 37 - 37 = -13 - 37.
Step 3: Simplify both sides to obtain: y = -50.
Final Answer: y = -50.

Solving an Equation using Addition/Subtraction

QUESTION

Determine if x = 3/2 is a solution to 4x - 2 = 2x + 1.

STEP-BY-STEP ANSWER:

Step 1: Substitute x = 3/2 into both sides of the equation.
Step 2: Calculate the left side: 4(3/2) - 2 = 6 - 2 = 4.
Step 3: Calculate the right side: 2(3/2) + 1 = 3 + 1 = 4.
Step 4: Since both sides equal 4, the solution is verified.
Final Answer: x = 3/2 is a valid solution.

Verifying a Solution

QUESTION

Solve the equation: 2x = 6

STEP-BY-STEP ANSWER:

Step 1: Identify the multiplication by 2.
Step 2: Use the Division Property of Equality by dividing both sides by 2: (2x)/2 = 6/2.
Step 3: Simplify to obtain: x = 3.
Final Answer: x = 3.

Solving an Equation with a Variable Multiplied by a Constant

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Common Mistakes

  • Neglecting to perform the same arithmetic operation on both sides of the equation.
  • Dropping or mismanaging negative signs during simplification.
  • Failing to simplify expressions completely before attempting to isolate the variable.
  • Misinterpreting phrases in word problems when translating them into equations.
  • Overlooking the need to check the final solution by substituting it back into the original equation.