Solved Exercises on Introduction to Variables in Grade 9

Master the concept of variables: using letters to represent unknowns, forming expressions, solving equations, and problem-solving through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Understanding Variables
Exercise 1
A number is doubled and then increased by 5. The result is 17. Find the number using a variable.
Definition:

Variable: A symbol (usually a letter) that represents an unknown number or value

Algebraic Expression: A mathematical phrase with numbers, variables, and operation symbols

Equation: A statement that two expressions are equal

Variable method:
  1. Identify the unknown value
  2. Assign a variable to represent the unknown
  3. Translate the problem into an algebraic equation
  4. Solve the equation
  5. Verify the solution
Unknown
Let x = the number
Expression
2x + 5
Equation
2x + 5 = 17
Step 1: Identify the unknown

We need to find a number, so let's call it x

Step 2: Translate the verbal expression

"A number is doubled" → 2x

"increased by 5" → + 5

"The result is 17" → = 17

Step 3: Form the equation

2x + 5 = 17

Step 4: Solve the equation

2x + 5 = 17

2x = 17 - 5

2x = 12

x = 6

Step 5: Verify the solution

Check: 2(6) + 5 = 12 + 5 = 17 ✓

x = 6
Final answer:

The number is 6

Applied rules:

• Variable assignment: Use a letter to represent the unknown

• Equation formation: Translate words into mathematical symbols

• Inverse operations: Subtract 5 from both sides, then divide by 2

2 Multiple Variables
Exercise 2
The sum of two numbers is 15. The larger number is 3 more than the smaller number. Find both numbers using variables.
Definition:

Multiple variables: Using different letters to represent different unknowns

System of equations: A set of equations with multiple unknowns

Substitution method: Expressing one variable in terms of another

Variables
x = smaller, y = larger
System
x + y = 15, y = x + 3
Solution
x = 6, y = 9
Step 1: Assign variables to both unknowns

Let x = the smaller number

Let y = the larger number

Step 2: Translate the conditions into equations

"The sum of two numbers is 15" → x + y = 15

"The larger number is 3 more than the smaller" → y = x + 3

Step 3: Substitute to solve

Substitute y = x + 3 into x + y = 15:

x + (x + 3) = 15

2x + 3 = 15

Step 4: Solve for x

2x = 15 - 3

2x = 12

x = 6

Step 5: Find y

y = x + 3 = 6 + 3 = 9

Step 6: Verify the solution

Check: 6 + 9 = 15 ✓, and 9 = 6 + 3 ✓

Smaller = 6, Larger = 9
Final answer:

The two numbers are 6 and 9

Applied rules:

• Multiple variables: Use different letters for different unknowns

• System of equations: Solve using substitution or elimination

• Verification: Check both conditions with the solution

3 Expressions with Variables
Exercise 3
If x = 4 and y = 7, evaluate the expression: 3x² - 2xy + y - 5.
Definition:

Evaluating expressions: Substituting values for variables and calculating

Order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction

Like terms: Terms with the same variable parts

Given
x = 4, y = 7
Expression
3x² - 2xy + y - 5
Evaluate
48 - 56 + 7 - 5
Step 1: Identify the values to substitute

Given: x = 4 and y = 7

Step 2: Substitute values into the expression

3x² - 2xy + y - 5

= 3(4)² - 2(4)(7) + 7 - 5

Step 3: Evaluate using order of operations

First, calculate exponents: 3(16) - 2(4)(7) + 7 - 5

Step 4: Perform multiplications

= 48 - 56 + 7 - 5

Step 5: Perform additions and subtractions from left to right

= (48 - 56) + 7 - 5

= -8 + 7 - 5

= -1 - 5

= -6

Result = -6
Final answer:

When x = 4 and y = 7, the expression 3x² - 2xy + y - 5 equals -6

Applied rules:

• Substitution: Replace each variable with its given value

• Order of operations: PEMDAS/BODMAS sequence

• Calculation: Perform arithmetic operations carefully

Variables Fundamentals
\(\text{Expression: } ax + b, \text{ Equation: } ax + b = c, \text{ Evaluation: } \text{substitute values}\)
Basic Variable Operations
Variable
x, y, z
Represents unknown
Expression
3x + 5
Has no equals sign
Equation
3x + 5 = 14
Has equals sign
Key definitions:

Variable: A letter that stands for an unknown number (often x, y, or z)

Constant: A fixed number in an expression

Coefficient: The number multiplied by a variable

Variable methodology:
  1. Identify unknowns: What are you trying to find?
  2. Assign variables: Choose appropriate letters
  3. Form expressions: Write mathematical phrases
  4. Create equations: Set up equalities
  5. Solve systematically: Use algebraic methods
Tip 1: Choose meaningful variable names when possible (e.g., 't' for time, 'd' for distance).
Tip 2: Always verify your solution by substituting back into the original problem.
Tip 3: Keep track of the order of operations when evaluating expressions.
Tip 4: When in doubt, draw a picture or write the problem in your own words.
Common errors: Forgetting to distribute, sign errors, calculation mistakes, not checking solutions.
Exam preparation: Practice translating word problems, memorize variable rules, work with multiple unknowns.
Essential rules:

• Variable assignment: Use any letter to represent unknowns

• Expression evaluation: Substitute values and follow order of operations

• Equation solving: Perform same operation on both sides

• Verification: Always check solutions in original problem

Solution: Exercises 4 to 5
4 Word Problem with Variables
Exercise 4
Sarah has twice as many apples as Tom. Together they have 30 apples. How many apples does each person have?
Definition:

Word problems: Real-world situations expressed mathematically

Relationships: Describing how quantities relate to each other

Translation: Converting words into mathematical expressions

Variables
T = Tom's apples, S = Sarah's apples
Relationship
S = 2T
Total
S + T = 30
Step 1: Identify what you need to find

We need to find the number of apples for both Sarah and Tom

Step 2: Choose variables

Let T = number of apples Tom has

Let S = number of apples Sarah has

Step 3: Translate the relationships

"Sarah has twice as many as Tom" → S = 2T

"Together they have 30" → S + T = 30

Step 4: Substitute and solve

Substitute S = 2T into S + T = 30:

2T + T = 30

3T = 30

T = 10

Step 5: Find Sarah's apples

S = 2T = 2(10) = 20

Step 6: Verify the solution

Tom has 10, Sarah has 20

Is Sarah's amount twice Tom's? 20 = 2(10) ✓

Do they total 30? 10 + 20 = 30 ✓

Tom = 10, Sarah = 20
Final answer:

Tom has 10 apples and Sarah has 20 apples

Applied rules:

• Problem translation: Convert words to mathematical relationships

• Substitution method: Replace one variable with its equivalent expression

• Verification: Check that the solution satisfies all conditions

5 Geometry with Variables
Exercise 5
The perimeter of a rectangle is 40 cm. The length is 4 cm more than the width. Find the dimensions of the rectangle using variables.
Definition:

Geometric formulas: Mathematical relationships in shapes

Rectangle perimeter: P = 2(length) + 2(width)

Dimension problems: Finding lengths and widths using variables

Variables
w = width, l = length
Relationship
l = w + 4
Perimeter
2l + 2w = 40
Step 1: Identify the geometric relationship

For a rectangle: Perimeter = 2(length) + 2(width)

Step 2: Assign variables to unknowns

Let w = width of the rectangle

Let l = length of the rectangle

Step 3: Translate the given information

"length is 4 cm more than width" → l = w + 4

"perimeter is 40 cm" → 2l + 2w = 40

Step 4: Substitute and solve

Substitute l = w + 4 into 2l + 2w = 40:

2(w + 4) + 2w = 40

2w + 8 + 2w = 40

4w + 8 = 40

4w = 32

w = 8

Step 5: Find the length

l = w + 4 = 8 + 4 = 12

Step 6: Verify the solution

Width = 8 cm, Length = 12 cm

Perimeter = 2(12) + 2(8) = 24 + 16 = 40 cm ✓

Length is 4 more than width? 12 = 8 + 4 ✓

Width = 8 cm, Length = 12 cm
Final answer:

The rectangle has width 8 cm and length 12 cm

Applied rules:

• Geometric formulas: Use correct formula for the shape

• Variable substitution: Replace variables with equivalent expressions

• Algebraic manipulation: Distribute and combine like terms

Detailed Summary: Introduction to Variables
\(\text{Expression: } ax + b, \text{ Equation: } ax + b = c, \text{ System: } \begin{cases} ax + by = c \\ dx + ey = f \end{cases}\)
Variable Expressions and Equations
Comprehensive definitions:

Variable: A symbol (usually a letter) that represents one or more numbers

Algebraic Expression: A combination of variables, numbers, and operations (e.g., 3x + 5)

Equation: A statement that two expressions are equal (e.g., 3x + 5 = 14)

Constant Term: A number in an expression that does not change (e.g., 5 in 3x + 5)

Coefficient: The number multiplied by a variable (e.g., 3 in 3x)

Complete variable methodology:
  1. Problem identification: Read carefully and identify what you need to find
  2. Variable assignment: Choose appropriate letters to represent unknowns
  3. Expression formation: Write mathematical expressions for given relationships
  4. Equation creation: Set up equations based on problem conditions
  5. Solution: Solve the equation(s) using algebraic methods
  6. Verification: Check that your solution makes sense in the original problem
Tip 1: Choose variable names that remind you what they represent (e.g., 'w' for width, 't' for time).
Tip 2: When solving equations, always perform the same operation on both sides to maintain balance.
Tip 3: For problems with multiple unknowns, use the substitution method or elimination method.
Tip 4: Always check your answer by substituting back into the original problem conditions.
Common applications: Word problems, geometry, physics, economics, everyday problem-solving.
Key skills: Translation, substitution, solving equations, verification, logical reasoning.
Essential rules and formulas:

• Variable substitution: Replace variables with their known values

• Order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction

• Equation solving: Perform inverse operations to isolate the variable

• System of equations: Use substitution or elimination to solve

• Verification: Always check solutions in original problem

Visualization: Variables and Expressions
Variable Relationships
Visualizing how variables relate to each other:
Linear relationships, substitution, and problem-solving
Examples of variable applications

Analysis: The chart shows different variable relationships and their applications.

  • Linear equations represent straight-line relationships
  • Variables can represent geometric dimensions
  • Systems of equations model multiple constraints

Questions & Answers

Question: How do I know which letter to use for a variable? Does it matter if I use x, y, or z?

Answer: You can use any letter as a variable, but here are some guidelines:

  • Traditional letters: x, y, z are commonly used for unknowns
  • Meaningful letters: Use letters that represent what you're finding (t=time, d=distance, w=width)
  • Consistency: If you use x for one unknown, use a different letter for another unknown
  • Context: In some subjects, letters have conventional meanings (t for time in physics)

The letter itself doesn't matter mathematically, but choosing meaningful letters helps you remember what each variable represents and makes problems easier to understand.

Question: What's the difference between an expression and an equation?

Answer: The key difference is the equals sign:

  • Expression: A mathematical phrase that doesn't have an equals sign (e.g., 3x + 5, 2y - 7, x²)
  • Equation: A mathematical sentence that has an equals sign showing two expressions are equal (e.g., 3x + 5 = 14, 2y - 7 = 9)

You can evaluate expressions (find their value when variables are known), but you solve equations (find the value of variables that make the equation true).

Expressions are like phrases, equations are like complete sentences.

Question: Why do we use variables? Can't we just solve problems with numbers?

Answer: Variables are essential for several important reasons:

  • Unknown values: We use variables when we don't know the value yet
  • Generalization: Variables allow us to write formulas that work for many different numbers
  • Patterns: Variables help us see relationships between quantities
  • Efficiency: Instead of solving the same type of problem repeatedly, we can solve it once with variables
  • Modeling: Variables help us create mathematical models of real-world situations

For example, the formula A = πr² works for any circle, not just one specific circle. Without variables, we'd have to rediscover this relationship every time we encountered a new circle.

Question: How do I know when to add, subtract, multiply, or divide when translating word problems?

Answer: Look for key words and phrases that indicate operations:

  • Addition: "sum," "total," "more than," "increased by," "combined"
  • Subtraction: "difference," "less than," "decreased by," "fewer than," "take away"
  • Multiplication: "product," "times," "of," "multiplied by," "twice," "double"
  • Division: "quotient," "divided by," "per," "out of," "ratio of"

However, be careful with "less than" - it reverses the order: "5 less than x" means x - 5, not 5 - x.

Always read the entire problem and think about the relationship between quantities to confirm your translation makes sense.

Question: What's the best way to check if my variable solution is correct?

Answer: The most effective method is substitution:

  • Substitute back: Put your solution back into the original equation(s)
  • Check both sides: Make sure both sides of the equation are equal
  • Verify conditions: Make sure your answer satisfies all conditions in the word problem
  • Reasonableness: Check if your answer makes sense in the context of the problem

For example, if you solve for someone's age and get a negative number, you know something is wrong. If your solution checks out in the equation but doesn't make sense in the real-world context, reconsider your setup.

Developing this habit of verification will save you many mistakes and build confidence in your algebra skills.