In Algebra, we move beyond just numbers to using symbols. These symbols help us solve problems where some quantities are unknown.
A symbol which can take various numerical values is called a variable (or literal). We usually use letters to represent them.
Definition: A quantity that can change.
Examples: $x, y, z, w, a, b, \dots$
A symbol which has a fixed value is called a constant. Its value does not change regardless of the problem.
Definition: A quantity that remains fixed.
Examples: $5, 3, -10, -\frac{1}{2}, 100, \dots$
One of the most important skills in algebra is translating verbal sentences into mathematical expressions. Below is a guide on how to write statements in symbolic form.
| Statement | Symbolic Form | Reasoning / Example |
| The sum of $a$ and $b$ | $a + b$ | Addition |
| The subtraction of $b$ from $a$ | $a – b$ | Note: “From $a$” means $a$ comes first. (Ex: Subtraction of 3 from 5 is $5 – 3 = 2$) |
| The product of 2 and $x$ | $2x$ | Multiplication sign is often omitted. (Ex: $3 \times x = 3x$) |
| Divide $x$ by 2 | $\frac{x}{2}$ | Division represented as a fraction. |
| 6 more than $x$ | $x + 6$ | “More than” implies addition. |
| $x$ decreased by 5 | $x – 5$ | “Decreased by” implies subtraction. |
| 5 times $x$ | $5x$ | Multiplication. |
| 40 divided by $x$ | $\frac{40}{x}$ | Numerator is 40, denominator is $x$. |
| 5 times $x$ added to 3 times $y$ | $3y + 5x$ | Combine multiplication and addition. |
| The subtraction of $2x$ from 3 | $3 – 2x$ | Again, “from 3” puts 3 first. |
| Half of $x$ | $\frac{x}{2}$ | Equivalent to dividing by 2. |
| One third of $x$ | $\frac{x}{3}$ | Equivalent to dividing by 3. |
| 4 less than half of $x$ | $\frac{x}{2} – 4$ | First find half of $x$, then subtract 4. |
When working with variables (like $a$), keep these properties in mind:
$1 \times a = a$
$-1 \times a = -a$
$\frac{a}{1} = a$
$\frac{a}{-1} = -a$
$0 \times a = 0$
An algebraic expression is made up of terms. A term can be:
A constant alone (e.g., $6$)
A variable alone (e.g., $x$)
A combination of variables and constants connected by multiplication or division.
Examples of Terms:
When variables and numbers are multiplied together to form a term, each distinct part is called a factor.
Example: In the term $5xy$:
It is written as $5 \times x \times y$
The factors are: 5, $x$, and $y$.
Example: In the number $6$:
$6 = 1 \times 6$ or $2 \times 3$
In a product containing two or more factors, any factor is called the co-efficient of the remaining factors.
Numerical Co-efficient: Usually refers to the number part.
Literal Co-efficient: Refers to the variable part.
Examples:
In the term $5x$:
$5$ is the co-efficient of $x$.
In the term $-6xy$:
$-6$ is the co-efficient of $xy$.
Special Cases (Hidden Co-efficients):
In $x$: The co-efficient is $1$.
Reason: $x = 1 \times x$
In $-x$: The co-efficient is $-1$.
Reason: $-x = -1 \times x$
In $\frac{x}{2}$: The co-efficient is $\frac{1}{2}$.
Reason: $\frac{x}{2} = \frac{1}{2} \times x$
To perform addition or subtraction in algebra, we must distinguish between like and unlike terms.
Terms that contain the exact same literal factors (same variables raised to the same powers) are called like terms.
Examples:
$2a$ and $3a$ are like terms.
$2a$ and $-3a$ are like terms.
$3pq$ and $4pq$ are like terms.
$-4xyz$ and $4xyz$ are like terms.
$\frac{1}{2}x$ and $\frac{3}{2}x$ are like terms.
Terms that have different variables or different powers of variables are unlike terms.
Examples:
$2a$ and $3b$ are unlike terms (different letters).
$3xy$ and $4pq$ are unlike terms.
$2$ and $a$ are unlike terms (one is constant, one is variable).
$\frac{3}{2}a$ and $\frac{3}{2}b$ are unlike terms.
You can only add like terms. To add them, you sum their numerical co-efficients and keep the variable same.
$2a + 3a = (2+3)a = \mathbf{5a}$
$5a + 6a + 7a = \mathbf{18a}$
$3b + 11b = \mathbf{14b}$
Grouping: $3a + 5a + 3b + 4b$
Combine $a$‘s: $3a + 5a = 8a$
Combine $b$‘s: $3b + 4b = 7b$
Result: $\mathbf{8a + 7b}$
Constants: $3a + 2 + 5a + 7$
Combine $a$‘s: $3a + 5a = 8a$
Combine constants: $2 + 7 = 9$
Result: $\mathbf{8a + 9}$
Similar to addition, you subtract the co-efficients of like terms.
$5a – 3a = \mathbf{2a}$
$-5a + 3a = \mathbf{-2a}$ (Because $-5 + 3 = -2$)
$-8m – 7m = \mathbf{-15m}$ (Because $-8 – 7 = -15$)
Practice what you have learned by solving these problems.
Part 1: Basic Concepts
The subtraction of $y$ from $p$ is __________
$-2 \times a =$ __________
$-1 \times p =$ __________
The factors of $5pq$ are __________
The co-efficient of $x$ in $\frac{x}{3}$ is __________
Part 2: Operations
$11a + 7a =$ __________
$-11a – 7a =$ __________
$-11a + 7a =$ __________
$11a – 7a =$ __________
$2a + 7b + 8a – 2b =$ __________
Part 1 Answers:
$p – y$ (Recall: “From $p$” puts $p$ first)
$-2a$
$-p$
$5, p, q$
$\frac{1}{3}$ (Because $\frac{x}{3} = \frac{1}{3} \times x$)
Part 2 Answers:
$18a$
$-18a$ (Since $-11 – 7 = -18$)
$-4a$ (Since $-11 + 7 = -4$)
$4a$
$10a + 5b$
Step-by-step: Group like terms: $(2a + 8a) + (7b – 2b) \rightarrow 10a + 5b$