CHAPTER 1:VARIABLES AND EXPRESSIONS 

1.1VARIABLES AND EXPRESSIONS 

Variables

Variables are symbols (usually letters) that represent unknown or changeable values in mathematical expressions. They act as placeholders for numbers that can vary.

Example: In the expression 2x + 5, 'x' is a variable.

Algebraic Expressions

Algebraic expressions combine numbers, variables, and mathematical operations to represent a value or relationship.

Examples:

• 3y + 2

• x² - 4x + 7

Translating Verbal to Algebraic Expressions

To convert a verbal expression to an algebraic one:

1. Identify the key terms and their mathematical meaning

2. Assign variables to unknown quantities

3. Use mathematical symbols to represent operations

Example:

Verbal: "Five more than twice a number"

Algebraic: 2x + 5 (where x represents "a number")

Translating Algebraic to Verbal Expressions

To convert an algebraic expression to a verbal one:

1. Identify the variables and their roles

2. Recognize the operations being performed

3. Use appropriate language to describe the mathematical relationships

Example:

Algebraic: 3y - 8

Verbal: "Three times a number, decreased by eight"

Common Phrases and Their Algebraic Equivalents

• "Sum of" → Addition (+)

• "Difference between" → Subtraction (-)

• "Product of" → Multiplication (×)

• "Quotient of" → Division (÷)

• "More than" → Addition (+)

• "Less than" → Subtraction (-)

Practice Examples 

1. Verbal to Algebraic:

   • "Six less than four times a number"

   • Solution: 4x - 6

2. Algebraic to Verbal:

   • 2(x + 3) - 5

   • Solution: "Twice the sum of a number and three, decreased by five"

Try this !!!

1. Write verbal expressions to represent each algebraic expression.

a).  n-7

b). 2a+6

c).  3x^4

d). 12a + 67b

2. Write algebraic expressions to represent each verbal expression.

a) Six less a number t

b)Three times a number plus 16.

c)20 divided by t to the fifth power .

d)The volume of a cylinder is times the radius r squared multiplied by the height h. Write an expression for the volume .

     3. Elsie buys a pizza for $16 and several bottles of water for $2 each. Let C represent the total    

         amount of money that Elsie spends and let w represent how many bottles of water she buys.

         Which equation best represents the situation?     (2pts)

     A.  C=2+16w

     B. C=16+2+w

     C.  C=16+2w

     D. C=2(w+16) 

1.2Order of operations 

Understanding Order of Operations 

The PEMDAS Rule

In algebra, we follow a specific order when solving mathematical expressions. This order is often remembered using the acronym PEMDAS:

1. Parentheses

2. Exponents

3. Multiplication and Division (left to right)

4. Addition and Subtraction (left to right)

Breaking Down PEMDAS

Parentheses

• Always solve what's inside parentheses first

• Example: In 3 + (4 × 2), solve 4 × 2 before adding 3

Exponents

• Next, calculate any exponents

• Example: In 3² + 4, solve 3² (which is 9) before adding 4

Multiplication and Division

• Perform these operations from left to right

• Example: In 12 ÷ 3 × 2, first do 12 ÷ 3, then multiply the result by 2

Addition and Subtraction

• Finally, perform addition and subtraction from left to right

• Example: In 10 - 5 + 2, first subtract 5 from 10, then add 2

Practical Application

Let's solve this expression: 2 + 3 × (4² - 5) ÷ 3 - 1

1. Parentheses: 4² - 5 = 16 - 5 = 11
   Now we have: 2 + 3 × 11 ÷ 3 - 1

2. Exponents: Already solved in step 1

3. Multiplication and Division (left to right):
   3 × 11 = 33
   33 ÷ 3 = 11
   Now we have: 2 + 11 - 1

4. Addition and Subtraction (left to right):
   2 + 11 = 13
   13 - 1 = 12

Final answer: 12

Common Mistakes to Avoid

• Don't solve from left to right ignoring PEMDAS

• Remember that multiplication and division have equal priority, as do addition and subtraction

• Always check your work to ensure you've followed the correct order

Try this!!!

1.3 Properties of Numbers 

Introduction to Algebraic Number Properties

In Algebra 1, we explore several important properties of numbers that help us solve equations and simplify expressions. These properties form the foundation for more advanced algebraic concepts.

Key Number Properties

1. Commutative Property

• Applies to addition and multiplication

• Example (Addition): 3 + 5 = 5 + 3

• Example (Multiplication): 2 × 4 = 4 × 2

2. Associative Property

• Applies to addition and multiplication

• Example (Addition): (2 + 3) + 4 = 2 + (3 + 4)

• Example (Multiplication): (2 × 3) × 4 = 2 × (3 × 4)

3. Distributive Property

• Combines multiplication with addition or subtraction

• Example: 3(x + 2) = 3x + 6

4. Identity Property

• Addition Identity: Adding 0 to any number doesn't change it (a + 0 = a)

• Multiplication Identity: Multiplying any number by 1 doesn't change it (a × 1 = a)

5. Inverse Property

• Additive Inverse: a + (-a) = 0

• Multiplicative Inverse: a × (1/a) = 1 (where a ≠ 0)

Practical Applications

Understanding these properties helps in:

1. Simplifying complex expressions

2. Solving equations more efficiently

3. Verifying mathematical proofs

Example Problem

Simplify: 3(x + 2) - 2(x - 1)

1. Use the distributive property:
   3x + 6 - 2x + 2

2. Combine like terms:
   x + 8

This problem demonstrates how understanding number properties can help simplify algebraic expressions.

Try this!!!

1.4 Understanding Relations 

What is a Relation?

In algebra, a relation is a connection between two sets of numbers or variables. It's like a rule that pairs elements from one set with elements from another set.

Types of Relations

1. One-to-One: Each element in the first set is paired with exactly one element in the second set.

2. One-to-Many: One element in the first set can be paired with multiple elements in the second set.

3. Many-to-One: Multiple elements in the first set can be paired with one element in the second set.

4. Many-to-Many: Multiple elements in the first set can be paired with multiple elements in the second set.

Ways to Represent Relations

1. Ordered Pairs: (x, y) where x is from the first set and y is from the second set.
   Example: {(1, 2), (3, 4), (5, 6)}

2. Mapping Diagrams: Visual representation using arrows to connect elements.

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3. Tables: Organizing the paired elements in columns.

input(x-value)           output(y-value)

5 6

7 4

9 1

4. Graphs: Plotting points on a coordinate plane.

    Example graphing these points  (1, 2), (3, 4), (5, 6)

5. Equations: Using algebraic expressions to define the relationship.
   Example: y = 2x + 1

Practical Applications

• Relationships between time and distance in physics

• Price and quantity in economics

• Age and height in biology

Key Concepts to Remember

• Domain: The set of all possible input values (x-values)

• Range: The set of all possible output values (y-values)

• Function: A special type of relation where each input has exactly one output

1.5 Functions

What is a Function?

A function is a special type of relation between two sets of numbers. In a function, each input (x-value) corresponds to exactly one output (y-value). Think of it as a machine that takes in a number and produces a unique result.

Determining Whether a Relation is a Function

To determine if a relation is a function, we use the vertical line test:

1. Plot the points of the relation on a coordinate plane.

2. Imagine drawing vertical lines through the graph.

3. If any vertical line intersects the graph more than once, it's not a function.

example 

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Examples:

1. Is y = x² a function?

   • Yes, because each x-value corresponds to only one y-value.

   • The graph is a parabola, and no vertical line will intersect it more than once.

2. Is y² = x a function?

   • No, because some x-values correspond to two y-values :+ ve value and negative value .

   • The graph is a sideways parabola, and vertical lines will intersect it twice.

Finding Function Values

To find function values, we substitute the given x-value into the function's equation and solve for y.

Example:

Let f(x) = 2x² - 3x + 1

Find f(2):

1. Replace x with 2 in the equation:
   f(2) = 2(2)² - 3(2) + 1

2. Simplify:
   f(2) = 2(4) - 6 + 1
   f(2) = 8 - 6 + 1
   f(2) = 3

So, when x = 2, the function value (y-value) is 3.

Key Points to Remember

• A function has one output for each input.

• Use the vertical line test to visually check if a graph represents a function.

• To find function values, substitute the given x-value into the function's equation.

Try this!!!

1.6 Interpreting Graphs of Functions

Understanding the Basics

When we look at the graph of a function, we can learn a lot about its behavior. Key aspects to consider are:

• Positive and negative regions

• Increasing and decreasing intervals

• Extreme points (maximums and minimums)

Positive and Negative Regions

• A function is positive when its graph is above the x-axis (y > 0)

• A function is negative when its graph is below the x-axis (y < 0)

• The points where the graph crosses the x-axis are called roots or zeros

Increasing and Decreasing Intervals

• A function is increasing when it goes up from left to right

• A function is decreasing when it goes down from left to right

• To find these intervals, look at the slope of the graph

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Extrema Points

• Maximum points are the highest points on the graph

• Minimum points are the lowest points on the graph

• These can be local (highest/lowest in a specific area) or global (highest/lowest for the entire function)

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TRY THIS !!!

Consider the graph of a parabola (y = x² - 4x + 3)

• It's positive when x < 1 or x > 3

• It's negative between x = 1 and x = 3

• It's decreasing from x = -∞ to x = 2

• It's increasing from x = 2 to x = +∞

• It has a minimum point at (2, -1)

CHAPTER 2:LINEAR EQUATIONS  

2.1 Writing equations

What are Equation Translations?

Equation translation is the process of converting mathematical equations into words and vice versa. This skill is crucial for interpreting real-world problems and expressing them mathematically.

Translating Words to Equations

1. Identify key mathematical operations:

   • "sum" or "total" → addition (+)

   • "difference" or "decrease" → subtraction (-)

   • "product" or "times" → multiplication (×)

   • "quotient" or "divided by" → division (÷)

2. Look for phrases indicating equality:

   • "is equal to"

   • "is the same as"

   • "results in"

3. Recognize variables:

   • "a number" → x, y, or n

   • "an unknown quantity" → any letter (commonly x, y, z)

Example:

"The sum of twice a number and five is equal to thirteen"

• Twice a number: 2x

• Sum with five: 2x + 5

• Equal to thirteen: = 13

• Resulting equation: 2x + 5 = 13

Translating Equations to Words

1. Start with the left side of the equation

2. Describe each term and operation

3. Use "is equal to" for the equals sign

4. Describe the right side of the equation

Example:

3x - 7 = 20

• "Three times a number"

• "minus seven"

• "is equal to"

• "twenty"

Full translation: "Three times a number minus seven is equal to twenty"

Practice Exercise

Try translating these:

1. Words to Equation: "The difference between a number and six is twelve"

2. Equation to Words: 4y + 2 = 30

Tips for Success

• Be precise with your language

• Practice regularly with various examples

• Visualize real-world scenarios for better understanding

TRY THIS!!!

1. Translate the equation y = 3x + 7 into a word problem.
   
   

2. Convert the following word problem into a linear equation: "A car rental company charges a flat fee of $30 plus $0.25 per mile driven."
   
   
   

3. Transform this statement into a linear equation: "The sum of three consecutive integers is 51."
   
   

4. Write a linear equation for: "A rectangle's length is 3 units more than twice its width."
   
   

5. Convert y = -4x + 10 into a word problem.
   
   

6. Translate this word problem into an equation: "The perimeter of a rectangle is 36 cm. Its length is 3 cm more than its width."
   
   

7. Express 5x + 2y = 20 as a word problem.
   
   

8. Write an equation for: "A plumber charges $75 for a call-out fee and $60 per hour of work."
   
   

9. Translate the word problem into an equation: "The difference between a number and its double is 15."
   
   

2.2 Solving One-Step Equations

What are One-Step Equations?

One-step equations are simple algebraic equations that can be solved in a single mathematical operation. They typically involve one variable and one operation (addition, subtraction, multiplication, or division).

The Concept of Balance

Think of an equation as a balance scale. The equals sign (=) represents the center of the scale, and both sides must always be equal or balanced.

Solving Process

1. Identify the operation being performed on the variable.

2. Perform the inverse operation on both sides of the equation to isolate the variable.

Examples

Addition/Subtraction

Equation: x + 5 = 12

Solution:

• Subtract 5 from both sides

• x = 12 - 5

• x = 7

Multiplication/Division

Equation: 3y = 21

Solution:

• Divide both sides by 3

• y = 21 ÷ 3

• y = 7

Tips for Success

• Always perform the same operation on both sides of the equation

• Check your answer by substituting it back into the original equation

• Practice with various types of one-step equations to build confidence

TRY THIS !!!

1. What is the value of x in the equation 3x = 15?
   
   

2. Solve for y: y - 7 = 12
   
   

3. In the equation 4z = 28, what is the value of z?
   
   

4. True or False: The solution to the equation a + 9 = 23 is a = 14.

5. Fill in the blank: In the equation 5b = 45, b = ___
   
   
   

6. Which of the following is the correct solution to 2x = 18?
   a) x = 7
   b) x = 8
   c) x = 9
   d) x = 10
   
   

7. Solve for m: m ÷ 4 = 6
   
   

8. What is the value of y in the equation y + 15 = 32?
   
   

9. Match the equation with its solution:

   1. 3x = 21 a) x = 8

   2. x - 5 = 3 b) x = 7

   3. x + 2 = 10 c) x = 8
      
      

10. Solve for k: -5k = 35
    
    

2.3 Solving Multi-Step Equations

What are Multi-Step Equations?

Multi-step equations are algebraic equations that require more than one operation to solve. They typically involve a combination of addition, subtraction, multiplication, and division.

The General Process

1. Simplify each side of the equation if necessary

2. Use inverse operations to isolate the variable

3. Solve for the variable

Key Concepts

• Order of operations (PEMDAS)

• Inverse operations

• Like terms

Solving Strategy

1. Combine like terms on each side of the equation

2. Use addition or subtraction to get variable terms on one side and constant terms on the other

3. Use multiplication or division to isolate the variable

Example

Let's solve: 3(x + 2) - 4 = 2x + 10

1. Distribute the 3: 3x + 6 - 4 = 2x + 10

2. Simplify: 3x + 2 = 2x + 10

3. Subtract 2x from both sides: x + 2 = 10

4. Subtract 2 from both sides: x = 8

Example

Let's solve the equation: 3x + 5 = 2x - 7

1. Simplify (already simplified in this case)

2. Move variables to one side:
   3x - 2x = -7 - 5

3. Combine like terms:
   x = -12

4. The variable is already isolated, so we're done!

Common Mistakes to Avoid

• Forgetting to distribute when there's a number outside parentheses

• Incorrectly applying the order of operations

• Not performing the same operation on both sides of the equation

Practice Tips

• Start with simpler equations and gradually increase complexity

• Check your answers by plugging the solution back into the original equation

• Use online resources or math apps for additional practice and instant feedback

TRY THIS!

1. Which of the following is the correct first step in solving the equation 3(x + 2) - 4 = 5?
   a) Subtract 4 from both sides
   b) Divide both sides by 3
   c) Add 4 to both sides
   d) Distribute the 3
   
   
   

2. Solve for x: 2(x - 3) + 4 = 10
   
   
   

3. What is the value of y in the equation 4y - 7 = 3y + 5?

4. True or False: In the equation 5(x + 2) = 3x + 16, x = 4
   
   
   

5. Which step is incorrect in solving 2(x + 3) - 5 = x + 1?
   Step 1: 2x + 6 - 5 = x + 1
   Step 2: 2x + 1 = x + 1
   Step 3: x = 0
   
   a) Step 1
   b) Step 2
   c) Step 3
   d) All steps are correct
   
   
   

6. Solve for a: 3(a - 2) + 4 = 2(a + 1)
   
   
   

7. What is the correct order of operations for solving 4(2x - 1) + 3 = 15?

   1. Distribute

   2. Subtract 3 from both sides

   3. Combine like terms

   4. Divide by 8

a) 1, 3, 2, 4
b) 1, 2, 3, 4
c) 2, 1, 3, 4
d) 3, 1, 2, 4

8. In the equation 5(x - 2) = 3(x + 4), what is the value of x?
   
   
   

9. Which of the following equations is equivalent to 3(x + 2) = 18?
   a) x + 2 = 6
   b) 3x + 6 = 18
   c) 3x + 2 = 18
   d) x + 6 = 18
   
   
   

10. Solve for y: 2(y - 3) + 4y = 3(y + 1) - 5

11. Multiple Choice: Which of the following equations has the solution x = 3?
    A) 2x + 5 = 3x - 2
    B) 4x - 7 = 3x + 4
    C) 5x + 2 = 4x + 5
    D) 3x - 6 = 2x + 3
    
    

12. True or False: The equation 7(x - 2) = 5(x + 1) has no solution.
    
    

13. Fill in the blank: In the equation 3(2x - 1) = 2(3x + 4), the value of x is _____.
    
    

14. Short Answer: Solve the equation: 5(x + 2) - 3(x - 1) = 2(x + 3)

15. Solve for x: 3(x + 2) - 5 = 2x + 7
    
    

16. Find the value of y: 2(y - 3) + 4y = 18
    Solve the equation: 5(2x - 1) = 3(x + 4) + 7
    
    

17. If the sum of three consecutive integers is 51, what is the smallest integer?
    
    

18. Solve for a: 4(a + 2) - 3(a - 1) = 2a + 5
    
    

19. Find the value of b: 2(3b - 1) - 5 = b + 7
    
    

20. The sum of four consecutive even integers is 52. What is the largest integer?
    
    

21. Solve the equation: 3(2x + 1) - 2(x - 3) = 4x + 5
    
    

22. If the product of three consecutive integers is 210, what is the middle integer?
    
    

23. Solve for y: 2(y + 3) - 3(y - 2) = 4y - 11
    
    

 2-4: Solving Equations with Variables on Both Sides

✅ Example

Problem: 5x−2=3x+65x - 2 = 3x + 65x−2=3x+6
Steps:
→ Subtract 3x: 2x−2=62x - 2 = 62x−2=6
→ Add 2: 2x=82x = 82x=8
→ Divide: x=4x = 4x=4

 Practice

1. 4x+1=2x+5

2. 6x−3=4x+7

3. 5x + 2 = 3x + 10

4. 5x+2=3x+10
   
   

5. 2x+8=2x−4
   
   

6. 3x+1=3x+5

🔹 2-5: Solving Literal Equations

✅ Example

Problem: Solve for h in A=1/2bh
Steps: Multiply both sides by 2: 2A=bh2
Then divide by b: h=Abh/2 

📝 Practice:

1. Solve d=rtd for t
   
   

2. Solve P=2l+2w for w
   
   

3. Solve y=mx+b for x
   
   

4. Solve A=lw for l
   
   

5. Solve C=2πr for r