IB MYP 3 Mathematics 3.1 Algebraic Expressions, Substitution and Like Terms Study Notes - New Syllabus
IB MYP 3 Mathematics 3.1 Algebraic Expressions, Substitution and Like Terms Study Notes
IB MYP 3 Mathematics 3.1 Algebraic Expressions, Substitution and Like Terms Study Notes at IITian Academy focus on specific topics and types of questions asked in the actual exam. Study Notes focus on the IB MYP 3 Mathematics syllabus with guiding questions of
Algebraic expression: A combination of numbers, variables and mathematical operations without an equals sign.
Term: A number, variable, or product of numbers and variables separated by \(+\) or \(-\) signs.
Variable: A letter or symbol representing a value that can change or is unknown.
Coefficient: The numerical factor multiplying a variable.
Constant: A number whose value does not change.
Substitution: Replacing a variable with a known value.
Like terms: Terms with exactly the same variables raised to the same powers.
Collecting like terms: Combining the coefficients of like terms to simplify an expression.
Simplifying: Combining like terms to write an equivalent expression in a simpler form.
Key rule: Only like terms can be added or subtracted, and negative values should be placed in brackets when substituted.
3.1 – Algebraic Expressions, Substitution and Like Terms
Algebra is a way of representing unknown or changing quantities using letters, numbers and mathematical operations. Algebraic expressions allow relationships and calculations to be written in a compact form.
Algebraic Expressions
An algebraic expression is a combination of numbers, variables and operations. An expression does not contain an equals sign.
Examples include:
- \(3x+5\)
- \(7a-2\)
- \(4x^2+3x-8\)
- \(\dfrac{x}{5}+2\)
An expression has no equals sign: \(3x+5\).
An equation contains an equals sign: \(3x+5=20\).
Terms
A term is a number, variable, or product of numbers and variables separated by \(+\) or \(-\) signs.
Consider:
\(5x+3y-7\)
The terms are:
- \(5x\)
- \(3y\)
- \(-7\)
The sign belongs to the term that follows it.
Example:
\(4x-6y+9\)
The terms are \(4x\), \(-6y\), and \(9\).
Variables
A variable is a letter or symbol used to represent a value that can change or is unknown.
Common variables include \(x\), \(y\), \(a\), \(b\), and \(n\).
For example, in:
\(7x+4\)
\(x\) is the variable.
Coefficients
A coefficient is the numerical factor multiplying a variable.
For example, in:
\(8x\)
the coefficient of \(x\) is \(8\).
In:
\(-3y\)
the coefficient of \(y\) is \(-3\).
If no number is written, the coefficient is \(1\) or \(-1\).
\(x=1x\)
\(-x=-1x\)
Constants
A constant is a number whose value does not change.
For example, in:
\(6x+11\)
\(6\) is the coefficient and \(11\) is the constant.
Parts of an Algebraic Expression
| Expression | Variable | Coefficient | Constant |
|---|---|---|---|
| \(5x+7\) | \(x\) | \(5\) | \(7\) |
| \(-3y+4\) | \(y\) | \(-3\) | \(4\) |
| \(2a^2-5a+9\) | \(a\) | \(2,-5\) | \(9\) |
Writing Algebraic Expressions
Words can be translated into algebraic expressions.
| Words | Expression |
|---|---|
| \(5\) more than \(x\) | \(x+5\) |
| \(7\) less than \(x\) | \(x-7\) |
| \(4\) times \(x\) | \(4x\) |
| \(x\) divided by \(3\) | \(\dfrac{x}{3}\) |
| Twice \(x\) plus \(9\) | \(2x+9\) |
“\(5\) less than \(x\)” means \(x-5\), not \(5-x\).
Substitution
Substitution means replacing a variable with a known value.
For example, evaluate:
\(3x+7\)
when \(x=5\).
Replace \(x\) with \(5\):
\(3(5)+7\)
\(=15+7\)
\(=22\)
Answer: \(22\)
Substitution with Negative Numbers
When substituting a negative number, use brackets.
Evaluate \(x^2+4x-3\) when \(x=-2\).
\((-2)^2+4(-2)-3\)
\(=4-8-3\)
\(=-7\)
Answer: \(-7\)
Always put a negative value inside brackets when substituting into an expression containing powers.
For example, if \(x=-3\):
\((-3)^2=9\), but \(-3^2=-9\).
Like Terms
Like terms have exactly the same variables raised to the same powers.
Examples of like terms:
- \(3x\) and \(7x\)
- \(5a^2\) and \(-2a^2\)
- \(4xy\) and \(9xy\)
- \(6\) and \(-3\)
Examples of unlike terms:
- \(3x\) and \(3y\)
- \(x\) and \(x^2\)
- \(2ab\) and \(2a\)
Collecting Like Terms
Only like terms can be added or subtracted.
For example:
\(5x+3x=8x\)
Another example:
\(7a-2a+5\)
\(=5a+5\)
With several terms:
\(4x+7-2x+5\)
\(=4x-2x+7+5\)
\(=2x+12\)
You cannot combine unlike terms.
\(3x+4y\) cannot be simplified further because \(x\) and \(y\) are different variables.
Simplifying Algebraic Expressions
To simplify an algebraic expression:
- Identify the terms.
- Group like terms.
- Add or subtract their coefficients.
- Write the simplified expression.
Example:
\(6x+4y-2x+3y-8\)
\(=6x-2x+4y+3y-8\)
\(=4x+7y-8\)
Answer: \(4x+7y-8\)
Variable: A letter representing an unknown or changing value.
Term: A number, variable, or product separated by \(+\) or \(-\).
Coefficient: The numerical factor multiplying a variable.
Constant: A number whose value does not change.
Like terms: Terms with exactly the same variables and powers.
Substitution: Replacing a variable with a known value.
Simplify: Combine like terms to write an equivalent expression in a simpler form.
Example 1:
Consider the expression:
\(4x^2-3x+7\)
a) State the variable, coefficients and constant.
b) Evaluate the expression when \(x=2\).
c) Evaluate the expression when \(x=-2\).
▶️ Answer/Explanation
Answer
a)
Variable: \(x\)
Coefficients: \(4\) and \(-3\)
Constant: \(7\)
b) When \(x=2\):
\(4(2)^2-3(2)+7\)
\(=16-6+7\)
\(=17\)
Answer: \(17\)
c) When \(x=-2\):
\(4(-2)^2-3(-2)+7\)
\(=16+6+7\)
\(=29\)
Answer: \(29\)
Example 2:
A rectangle has length \(x+5\) cm and width \(3x-2\) cm.
a) Write an expression for the perimeter of the rectangle.
b) Simplify your expression.
c) Find the perimeter when \(x=4\).
▶️ Answer/Explanation
Answer
a) Write the perimeter expression.
The perimeter of a rectangle is:
\(P=2(\text{length})+2(\text{width})\)
Therefore:
\(P=2(x+5)+2(3x-2)\)
b) Simplify.
\(P=2x+10+6x-4\)
\(P=8x+6\)
Answer: \(P=8x+6\) cm.
c) When \(x=4\):
\(P=8(4)+6\)
\(=38\)
Answer: \(38\) cm.