Lesson 1: Algebra Review - Building the Foundation

Study plan: Six short concept blocks of 10-15 minutes each, plus practice.

This lesson begins with the language of algebra and ends with simple linear equations. No previous algebra knowledge is assumed. Study one concept block, answer its checkpoint, and pause before starting the next block.

1. Learning Objectives

By the end of this lesson, you should be able to:

• recognize numbers, variables, constants, coefficients, and operators;
• evaluate an expression using the correct order of operations;
• simplify expressions by combining like terms;
• use the distributive property; and
• solve and verify a simple linear equation.

2. Intuition

Algebra is arithmetic with labelled boxes.

Imagine a shopping bill:

3 notebooks at x rupees each + a 20-rupee pen

Total cost = 3x + 20

The letter x is not mysterious. It is simply a placeholder for a number that may be unknown or may change. If one notebook costs 40 rupees, replace x with 40:

3x + 20 = 3(40) + 20 = 140

This ability to describe a whole family of calculations with one expression is why algebra is useful in programming, science, finance, and computing.

3. Theory

Concept 1: The language of algebra

An expression is a mathematical phrase made from numbers, letters, and operations. For example:

4x + 7

Its parts are:

Part	Name	Meaning
x	Variable	A number that can vary or is not yet known
4 in 4x	Coefficient	The number multiplying the variable
7	Constant	A fixed number
4x, 7	Terms	Parts separated by + or -
+	Operator	An instruction to perform an operation

Multiplication is often written without the multiplication sign:

4 * x = 4x
a * b = ab

Simple example

In 5y - 9, the variable is y, its coefficient is 5, and the constant is -9.

Real-world application

If a taxi charges 50 rupees initially and 15 rupees per kilometre, the cost for k kilometres is:

Cost = 15k + 50

University-style problem

Identify the terms, variables, coefficients, and constant in:

6a - 3b + 11

Step 1: Terms are separated by addition or subtraction.

Terms: 6a, -3b, 11

Step 2: Letters represent variables.

Variables: a, b

Step 3: The number multiplying each variable is its coefficient.

Coefficient of a: 6
Coefficient of b: -3

Step 4: The number without a variable is the constant.

Constant: 11

Checkpoint 1

In 8p - 5, identify the variable, coefficient, and constant. Write your answer before continuing.

Concept 2: Substitution and evaluating expressions

To evaluate an expression, replace each variable with its given value and calculate the result. This replacement is called substitution.

Simple example

Evaluate 2x + 3 when x = 4.

2x + 3
= 2(4) + 3
= 8 + 3
= 11

Parentheses around the substituted value make multiplication clear.

Real-world application

Computer programs use variables in exactly this way:

price = 40
total = 3 * price + 20
print(total)  # 140

The algebraic variable becomes a memory location in a program.

University-style problem

Evaluate 3x^2 - 2y + 5 for x = 2 and y = -1.

Step 1: Substitute both values carefully.

3(2)^2 - 2(-1) + 5

Step 2: Calculate the power first.

3(4) - 2(-1) + 5

Step 3: Perform multiplication.

12 + 2 + 5

Notice that -2(-1) = +2 because a negative multiplied by a negative is positive.

Step 4: Add.

Answer = 19

Checkpoint 2

Evaluate 4m - 7 when m = 3. Write your answer before continuing.

Concept 3: Order of operations

When an expression contains several operations, everyone must calculate in the same order. Use BODMAS:

B  Brackets
O  Orders: powers and roots
D  Division       (left to right)
M  Multiplication (left to right)
A  Addition       (left to right)
S  Subtraction    (left to right)

Multiplication does not always come before division; they have equal priority and are handled from left to right. The same is true for addition and subtraction.

Visual explanation

          2 + 3 * (4 + 1)^2
                    |
Step 1: brackets -> 2 + 3 * 5^2
                         |
Step 2: power    -> 2 + 3 * 25
                             |
Step 3: multiply -> 2 + 75
                         |
Step 4: add       -> 77

Simple example

10 - 2 * 3 = 10 - 6 = 4

Doing subtraction first would incorrectly give 24.

Real-world application

Spreadsheets, calculators, and programming languages use precedence rules. A formula such as base + rate * hours multiplies before adding unless parentheses explicitly change the order.

University-style problem

Evaluate:

18 / 3 * 2 + (5 - 1)^2

Step 1: Brackets.

18 / 3 * 2 + 4^2

Step 2: Power.

18 / 3 * 2 + 16

Step 3: Division and multiplication from left to right.

6 * 2 + 16
= 12 + 16

Step 4: Addition.

Answer = 28

Checkpoint 3

Evaluate 6 + 2 * (7 - 4)^2. Write your answer before continuing.

Concept 4: Like terms and simplification

Like terms have exactly the same variable part. They can be combined just as similar objects can be counted together.

3 apples + 2 apples = 5 apples
3x      + 2x       = 5x

But apples and oranges are different categories:

3x + 2y cannot be simplified to 5xy or 5x

The terms 4x and 7x are like terms. The terms 4x, 4x^2, and 4y are not like terms.

Simple example

5x + 3 - 2x + 4
= (5x - 2x) + (3 + 4)
= 3x + 7

Real-world application

Symbolic mathematics software and compilers simplify repeated operations to reduce computation. For example, 2 * size + 5 * size can be optimized to 7 * size.

University-style problem

Simplify 7a - 3b + 2a + 5b - 4.

Step 1: Group like terms.

(7a + 2a) + (-3b + 5b) - 4

Step 2: Add their coefficients.

9a + 2b - 4

Answer: 9a + 2b - 4.

Checkpoint 4

Simplify 6x + 4 - 2x + 9. Write your answer before continuing.

Concept 5: The distributive property

The distributive property means multiplying every item inside a bracket.

Imagine three gift bags, each containing one pen and two chocolates:

3 bags * (1 pen + 2 chocolates)
= 3 pens + 6 chocolates

In algebra:

a(b + c) = ab + ac
a(b - c) = ab - ac

Visual explanation

      3(x + 4)
      /       \
   3 * x     3 * 4
      \       /
       3x + 12

Simple example

2(x + 5) = 2x + 10

Real-world application

If five students each buy a notebook costing n rupees and a pen costing 10 rupees, the total can be written in two equivalent ways:

5(n + 10) = 5n + 50

University-style problem

Simplify 4(2x - 3) + 5x.

Step 1: Distribute 4 to both terms in the bracket.

8x - 12 + 5x

Step 2: Combine like terms.

13x - 12

Answer: 13x - 12.

Checkpoint 5

Simplify 3(y + 2) + 2y. Write your answer before continuing.

Concept 6: Solving linear equations

An equation states that two expressions have the same value. Think of it as a balanced weighing scale:

          2x + 3 = 11

       left side     right side
           \           /
            \_________/
              balance

Whatever operation is performed on one side must also be performed on the other side. The goal is to leave the variable alone.

Formula explanation

For the general linear equation:

ax + b = c, where a is not 0

subtract b from both sides and then divide both sides by a:

ax + b = c
ax     = c - b
x      = (c - b) / a

The formula is simply a summary of balance-preserving steps.

Simple example

Solve x + 5 = 9.

x + 5 - 5 = 9 - 5
x = 4

Real-world application

Suppose a mobile plan charges a fixed 100 rupees plus 2 rupees per minute. A bill is 220 rupees. The minutes used satisfy:

100 + 2m = 220

Solving the equation finds m = 60 minutes.

University-style problem

Solve and verify 3(x - 2) + 4 = 19.

Step 1: Open the bracket.

3x - 6 + 4 = 19

Step 2: Combine constants.

3x - 2 = 19

Step 3: Add 2 to both sides.

3x = 21

Step 4: Divide both sides by 3.

x = 7

Step 5: Verify by substituting x = 7 into the original equation.

Left side = 3(7 - 2) + 4
          = 3(5) + 4
          = 19

Right side = 19

Both sides match, so x = 7 is correct.

Checkpoint 6

Solve and verify 2x + 5 = 17. Do not continue until you have written the value of x and checked it in the original equation.

4. Visual Explanation

The ideas in this lesson form one small pipeline:

Real situation
     |
     v
Variable and expression      3x + 20
     |
     v
Substitute or simplify       3(40) + 20
     |
     v
Use operation rules          120 + 20
     |
     v
Result                       140

If two expressions are equal:

Equation -> preserve balance -> isolate variable -> verify answer

5. Formula Explanation

These are the only rules to remember from this lesson:

Rule	Meaning
a + a + a = 3a	Repeated addition becomes multiplication
ax + bx = (a + b)x	Combine coefficients of like terms
a(b + c) = ab + ac	Multiply every term inside the bracket
a(b - c) = ab - ac	Distribute across subtraction too
ax + b = c gives x = (c - b)/a	Undo addition, then multiplication

Do not memorize a rule without connecting it to its meaning. For example, a(b + c) distributes because there are a copies of everything inside the bracket.

6. Worked Examples

Mixed example

Evaluate 2(3x + 1) - x when x = 4.

Method 1: substitute first

2(3(4) + 1) - 4
= 2(12 + 1) - 4
= 26 - 4
= 22

Method 2: simplify first

2(3x + 1) - x
= 6x + 2 - x
= 5x + 2

When x = 4:
5(4) + 2 = 22

Both methods agree. Simplifying first is often faster when the expression must be evaluated for many different values.

7. Programming Connection

Algebra and programming use variables differently but relatedly:

Mathematics	Programming
x = 5 states that x has value 5	x = 5 assigns 5 to a memory location
2x means multiplication	Most languages require 2 * x
x^2 may mean x squared	Python uses x ** 2; many languages use x * x
An equation describes equality	Many languages test equality with ==

Example Python function:

def taxi_fare(kilometres):
    return 15 * kilometres + 50

print(taxi_fare(10))  # 200

Algebra helps design the formula; programming repeats it accurately for many inputs.

8. Real-Life Applications

• Chemistry: rearranging concentration and dilution equations.
• Programming: calculating memory, time complexity, and coordinates.
• Data science: expressing relationships between variables.
• Computer graphics: scaling and moving points.
• Finance: calculating cost, tax, interest, and profit.
• AI: defining model inputs, weighted sums, and loss functions.
• Algorithms: describing running time using variables such as n.

9. Summary

Key takeaways

• A variable is a placeholder for a number.
• An expression describes a calculation; an equation states that two expressions are equal.
• Substitute values using parentheses.
• Follow BODMAS and work left to right between equal-priority operations.
• Combine only like terms.
• Distribute multiplication to every term inside a bracket.
• Preserve equality by doing the same operation to both sides.
• Verify a solution by substituting it into the original equation.

Common mistakes

• Treating 3x as 3 + x instead of 3 * x.
• Combining unlike terms, such as writing 2x + 3 = 5x.
• Forgetting to multiply every term: 2(x + 3) is 2x + 6, not 2x + 3.
• Ignoring BODMAS.
• Losing a negative sign during substitution or simplification.
• Changing only one side of an equation.
• Stopping after finding x without checking the answer.

Quick revision notes

Expression:  3x + 5
Equation:    3x + 5 = 20
Like terms:  3x + 2x = 5x
Distribute:  3(x + 2) = 3x + 6
Solve:       3x + 5 = 20 -> 3x = 15 -> x = 5
Check:       3(5) + 5 = 20

10. Practice Questions

Complete these without looking back at the worked examples.

1. Easy: In 7x - 4, identify the variable, coefficient, and constant.
2. Easy: Evaluate 3a + 2 when a = 5.
3. Medium: Simplify 8x + 3 - 5x + 7.
4. Medium: Simplify 4(2y - 3) + y.
5. Hard: Solve and verify 5(2x - 1) + 3 = 38.

11. Mini Quiz

Answer all five questions before checking any notes:

1. What is the difference between an expression and an equation?
2. Evaluate 20 / 5 * 2 + 3.
3. Can 4x + 3y be simplified to 7xy? Explain why or why not.
4. Expand and simplify 2(3x + 4) - x.
5. A service charges a fixed fee of 80 rupees and 12 rupees per hour. The total charge is 200 rupees. Form an equation and find the number of hours.

Stop Here

Do not begin another topic yet. Complete the six checkpoints, five practice questions, and mini quiz. Continue only after the learner requests Next Lesson.