📖 Algebraic Technique F – Substitution

Algebraic Technique F

Substitution into an algebraic expression

📐 What Is an Algebraic Expression?

An algebraic expression is an expression containing variables (letters) and numbers.

Examples of expressions:

\(3x + 5\)
\(2a - b\)
\(x^2 + 2x + 1\)
\(\frac{a + b}{2}\)

🔄 What Is Substitution?

Substitution = replacing a variable (letter) with a given number and evaluating the expression.

💡 How to substitute:

Identify the variable (letter)
Replace it with the given value
Evaluate following the order of operations

✏️ Basic Examples

Example 1: evaluate \(3x + 2\) for \(x = 4\)

Substitute \(x = 4\):

\(3 \cdot 4 + 2 = 12 + 2 = 14\)

Answer: 14

Example 2: evaluate \(5x - 7\) for \(x = 3\)

\(5 \cdot 3 - 7 = 15 - 7 = 8\)

Answer: 8

Example 3: evaluate \(x^2 + 3\) for \(x = 5\)

\(5^2 + 3 = 25 + 3 = 28\)

Answer: 28

⚠️ Substituting a Negative Number

When substituting a negative number – place it in brackets!

Example 4: evaluate \(2x + 5\) for \(x = -3\)

Substitute with brackets:

\(2 \cdot (-3) + 5 = -6 + 5 = -1\)

Answer: -1

Example 5: evaluate \(x^2 - 4\) for \(x = -2\)

\((-2)^2 - 4 = 4 - 4 = 0\)

Note: \((-2)^2 = 4\) (minus times minus equals plus)

Answer: 0

🔢 Substitution in a Two-Variable Expression

Example 6: evaluate \(2a + 3b\) for \(a = 4\) and \(b = 2\)

\(2 \cdot 4 + 3 \cdot 2 = 8 + 6 = 14\)

Answer: 14

Example 7: evaluate \(a^2 - b^2\) for \(a = 5\) and \(b = 3\)

\(5^2 - 3^2 = 25 - 9 = 16\)

Answer: 16

Example 8: evaluate \(\frac{x + y}{2}\) for \(x = 10\) and \(y = 6\)

\(\frac{10 + 6}{2} = \frac{16}{2} = 8\)

Answer: 8

🎯 Applications of Substitution

1. Verifying an equation solution:

To check whether \(x = 3\) solves \(2x + 1 = 7\):

Substitute: \(2 \cdot 3 + 1 = 6 + 1 = 7\) ✓

Yes, \(x = 3\) is a solution!

2. Evaluating a function:

For \(f(x) = x^2 + 1\), find \(f(3)\):

\(f(3) = 3^2 + 1 = 9 + 1 = 10\)

3. Real-world formulas:

Perimeter of a rectangle: \(P = 2a + 2b\)

If \(a = 5\) and \(b = 3\):

\(P = 2 \cdot 5 + 2 \cdot 3 = 10 + 6 = 16\)

💡 Tips for the Exam

Negative number? Use brackets!

Remember: order of operations

Note: \((-3)^2 \neq -3^2\)

📝 Summary

Substitution = replacing a variable with a number

Negative number → use brackets!

Evaluate following the order of operations

Worked Examples

🍎 In the fruit basket:
The basket has 3 apples and 5 more apples.
Simplify the expression: \(3a + 5a\)

Show solution

\(8a\)

✓ Correct

\(15a\)

\(3a+5a\) (No simplification)

\(8a^2\)

💡 Explanation — What are like terms?
🍎 Using apples as an example:
3 apples + 5 apples = 8 apples
📐 In algebra:
\(3a + 5a\)
The two terms share the same variable (a) — they are like terms.
🔢 How do we simplify?
Add the coefficients: \(3 + 5 = 8\)
The variable stays: \(a\)
Answer: \(8a\)
✨ Golden rule: Like terms share exactly the same variable!
You can combine them by adding or subtracting their coefficients.

🍌🍊 Mixed basket:
The basket has 4 bananas and 3 oranges.
Can this expression be simplified? \(4b + 3o\)?

Show solution

Cannot be simplified — different terms

✓ Correct

\(7bo\)

\(7b\)

\(7o\)

💡 Explanation — Why cannot this be simplified?
🍌🍊 Using fruit as an example:
4 bananas + 3 oranges = 4 bananas and 3 oranges.
You cannot say "7 somethings" — they are not the same thing!
📐 In algebra:
\(4b + 3o\)
b and o are different variables — these are unlike terms.
🚫 Important rule:
You can only combine terms with exactly the same variable!
✅ Like terms: \(3x + 5x\) ← same variable
❌ Unlike terms: \(3x + 5y\) ← different variables
💭 Think: you cannot add apples and bananas into one number!

🥕 Carrots in the garden:
We picked 7 carrots and ate 3 carrots.
Simplify the expression: \(7c - 3c\)

Show solution

\(4c\)

✓ Correct

\(10c\)

\(21c\)

\(4\)

💡 Explanation — Subtracting like terms
🥕 Using carrots as an example:
7 carrots − 3 carrots = 4 carrots
📐 In algebra:
\(7c - 3c\)
Both terms share the same variable (c) — they are like terms.
🔢 How do we subtract like terms?
Subtract the coefficients: \(7 - 3 = 4\)
The variable stays: \(c\)
Answer: \(4c\)
✨ Rule for subtraction:
Just like addition — subtract the coefficients!
\(7c - 3c = (7-3)c = 4c\)
💭 Remember: always check that the variable is identical before simplifying!

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