Understanding Variables and Expressions: 100 Simplified Examples in Algebra

Table of Contents

Variables and expressions are fundamental concepts in algebra and mathematics in general. Variables are symbols, often letters, that represent numbers. They allow us to create general rules or formulas that apply to many different situations. Expressions are combinations of variables, numbers, and mathematical operations such as addition, subtraction, multiplication, and division.

In this topic, we will explore variables and expressions in detail and go through 100 examples to clarify their usage.

1. What is a Variable?

A variable is a symbol (usually a letter) that represents a number that can change or vary. In algebra, common variables are $x$ , $y$ , and $z$ . Variables can represent unknown values in equations or be used to express general mathematical principles.

For example:

In the equation $x + 3 = 5$ , the variable $x$ represents an unknown number.
In the formula for the area of a circle, $A = π r^{2}$ , $r$ is the variable representing the radius of the circle.

2. What is an Expression?

An expression is a mathematical phrase that can include numbers, variables, and operations. It does not have an equal sign like an equation. Expressions can be simplified, evaluated, or used to create equations.

For example:

$3 x + 2$ is an algebraic expression.
$5 a - 7 b + 12$ is another example of an expression.

Expressions can be as simple as a single number or variable or as complex as a combination of multiple terms.

3. Simplifying Expressions

To simplify an expression means to combine like terms or reduce it to its simplest form. Like terms are terms that have the same variables raised to the same power. For instance, in the expression $5 x + 3 x$ , both terms are “like terms” because they both contain the variable $x$ .

Example of Simplifying: $5 x + 3 x = 8 x$

In this case, the terms $5 x$ and $3 x$ are added together to form $8 x$ .

4. Evaluating Expressions

To evaluate an expression means to find its value for specific values of the variables. This involves substituting numbers for variables and performing the necessary operations.

Example of Evaluating: If $x = 2$ , evaluate the expression $4 x + 7$ .

Solution: Substitute $2$ for $x$ in the expression: $4 (2) + 7 = 8 + 7 = 15$

5. Combining Variables and Constants

An expression can have both variables and constants (fixed numbers). For example, $3 x + 7$ contains the variable $x$ and the constant $7$ .

You can add, subtract, multiply, and divide variables just like numbers, but you must follow the rules of algebra.

6. 100 Examples of Variables and Expressions

Let’s explore 100 examples to understand variables and expressions better.

Examples 1 to 10

Example 1: Simplify $3 x + 5 x$ .

Solution:
$= (3 + 5) x$
$= 8 x$

Example 2: Evaluate $2 x + 4$ when $x = 3$ .

Solution:
Substitute $x = 3$ :
$2 (3) + 4 = 6 + 4$
$= 10$

Example 3: Simplify $7 a - 3 a$ .

Solution:
$= (7 - 3) a$
$= 4 a$

Example 4: Evaluate $5 x - 2$ when $x = 4$ .

Solution:
Substitute $x = 4$ :
$5 (4) - 2 = 20 - 2$
$= 18$

Example 5: Simplify $2 y + 7 y - 3 y$ .

Solution:
$= (2 + 7 - 3) y$
$= 6 y$

Example 6: Evaluate $6 x^{2} + 3 x$ when $x = 2$ .

Solution:
Substitute $x = 2$ :
$6 (2)^{2} + 3 (2)$
$= 6 (4) + 6$
$= 24 + 6$
$= 30$

Example 7: Simplify $4 b + 9 - b$ .

Solution:
$= (4 - 1) b + 9$
$= 3 b + 9$

Example 8: Evaluate $8 x + 10$ when $x = 5$ .

Solution:
Substitute $x = 5$ :
$8 (5) + 10$
$= 40 + 10$
$= 50$

Example 9: Simplify $12 y + 5 - 2 y$ .

Solution:
$= (12 - 2) y + 5$
$= 10 y + 5$

Example 10: Evaluate $7 x - 3$ when $x = 6$ .

Solution:
Substitute $x = 6$ :
$7 (6) - 3$
$= 42 - 3$
$= 39$

Examples 11 to 20

Example 11: Simplify $5 a + 4 a + 9$ .

Solution:
$= (5 + 4) a + 9$
$= 9 a + 9$

Example 12: Evaluate $3 x^{2} + 4 x$ when $x = 2$ .

Solution:
Substitute $x = 2$ :
$3 (2)^{2} + 4 (2)$
$= 3 (4) + 8$
$= 12 + 8$
$= 20$

Example 13: Simplify $6 p - 3 p + 4$ .

Solution:
$= (6 - 3) p + 4$
$= 3 p + 4$

Example 14: Evaluate $9 x + 2$ when $x = 1$ .

Solution:
Substitute $x = 1$ :
$9 (1) + 2$
$= 9 + 2$
$= 11$

Example 15: Simplify $8 b + 3 - 5 b$ .

Solution:
$= (8 - 5) b + 3$
$= 3 b + 3$

Example 16: Evaluate $2 x^{2} - 4 x$ when $x = 3$ .

Solution:
Substitute $x = 3$ :
$2 (3)^{2} - 4 (3)$
$= 2 (9) - 12$
$= 18 - 12$
$= 6$

Example 17: Simplify $10 a + 5 a - 6$ .

Solution:
$= (10 + 5) a - 6$
$= 15 a - 6$

Example 18: Evaluate $7 x + 12$ when $x = 4$ .

Solution:
Substitute $x = 4$ :
$7 (4) + 12$
$= 28 + 12$
$= 40$

Example 19: Simplify $9 p - 2 p + 11$ .

Solution:
$= (9 - 2) p + 11$
$= 7 p + 11$

Example 20: Evaluate $3 x^{2} + 5 x$ when $x = 2$ .

Solution:
Substitute $x = 2$ :
$3 (2)^{2} + 5 (2)$
$= 3 (4) + 10$
$= 12 + 10$
$= 22$

Examples 21 to 30

Example 21: Simplify $5 x + 2 x + 4$ .

Solution:
$= (5 + 2) x + 4$
$= 7 x + 4$

Example 22: Evaluate $4 x - 3$ when $x = 7$ .

Solution:
Substitute $x = 7$ :
$4 (7) - 3$
$= 28 - 3$
$= 25$

Example 23: Simplify $12 y + 3 y - 7$ .

Solution:
$= (12 + 3) y - 7$
$= 15 y - 7$

Example 24: Evaluate $8 x^{2} + 5 x$ when $x = 3$ .

Solution:
Substitute $x = 3$ :
$8 (3)^{2} + 5 (3)$
$= 8 (9) + 15$
$= 72 + 15$
$= 87$

Example 25: Simplify $4 a + 7 - 3 a$ .

Solution:
$= (4 - 3) a + 7$
$= a + 7$

Example 26: Evaluate $3 x + 8$ when $x = 5$ .

Solution:
Substitute $x = 5$ :
$3 (5) + 8$
$= 15 + 8$
$= 23$

Example 27: Simplify $6 y - 2 y + 5$ .

Solution:
$= (6 - 2) y + 5$
$= 4 y + 5$

Example 28: Evaluate $2 x^{2} + 3 x$ when $x = 1$ .

Solution:
Substitute $x = 1$ :
$2 (1)^{2} + 3 (1)$
$= 2 (1) + 3$
$= 2 + 3$
$= 5$

Example 29: Simplify $7 p + 4 - 5 p$ .

Solution:
$= (7 - 5) p + 4$
$= 2 p + 4$

Example 30: Evaluate $9 x + 6$ when $x = 2$ .

Solution:
Substitute $x = 2$ :
$9 (2) + 6$
$= 18 + 6$
$= 24$

Examples 31 to 40

Example 31: Simplify $3 x + 6 x + 9$ .

Solution:
$= (3 + 6) x + 9$
$= 9 x + 9$

Example 32: Evaluate $6 x - 4$ when $x = 4$ .

Solution:
Substitute $x = 4$ :
$6 (4) - 4$
$= 24 - 4$
$= 20$

Example 33: Simplify $10 y - 7 y + 3$ .

Solution:
$= (10 - 7) y + 3$
$= 3 y + 3$

Example 34: Evaluate $5 x^{2} + 2 x$ when $x = 3$ .

Solution:
Substitute $x = 3$ :
$5 (3)^{2} + 2 (3)$
$= 5 (9) + 6$
$= 45 + 6$
$= 51$

Example 35: Simplify $4 a + 8 - 2 a$ .

Solution:
$= (4 - 2) a + 8$
$= 2 a + 8$

Example 36: Evaluate $7 x + 9$ when $x = 5$ .

Solution:
Substitute $x = 5$ :
$7 (5) + 9$
$= 35 + 9$
$= 44$

Example 37: Simplify $9 y + 6 y - 4$ .

Solution:
$= (9 + 6) y - 4$
$= 15 y - 4$

Example 38: Evaluate $2 x^{2} + 6 x$ when $x = 4$ .

Solution:
Substitute $x = 4$ :
$2 (4)^{2} + 6 (4)$
$= 2 (16) + 24$
$= 32 + 24$
$= 56$

Example 39: Simplify $8 p + 3 - 4 p$ .

Solution:
$= (8 - 4) p + 3$
$= 4 p + 3$

Example 40: Evaluate $10 x + 5$ when $x = 3$ .

Solution:
Substitute $x = 3$ :
$10 (3) + 5$
$= 30 + 5$
$= 35$

Examples 41 to 50

Example 41: Simplify $7 x + 3 x + 5$ .

Solution:
$= (7 + 3) x + 5$
$= 10 x + 5$

Example 42: Evaluate $8 x - 6$ when $x = 4$ .

Solution:
Substitute $x = 4$ :
$8 (4) - 6$
$= 32 - 6$
$= 26$

Example 43: Simplify $12 y + 2 y - 9$ .

Solution:
$= (12 + 2) y - 9$
$= 14 y - 9$

Example 44: Evaluate $3 x^{2} + 4 x$ when $x = 5$ .

Solution:
Substitute $x = 5$ :
$3 (5)^{2} + 4 (5)$
$= 3 (25) + 20$
$= 75 + 20$
$= 95$

Example 45: Simplify $6 a + 4 - 5 a$ .

Solution:
$= (6 - 5) a + 4$
$= a + 4$

Example 46: Evaluate $9 x + 3$ when $x = 7$ .

Solution:
Substitute $x = 7$ :
$9 (7) + 3$
$= 63 + 3$
$= 66$

Example 47: Simplify $7 y - 4 y + 12$ .

Solution:
$= (7 - 4) y + 12$
$= 3 y + 12$

Example 48: Evaluate $6 x^{2} - 5 x$ when $x = 2$ .

Solution:
Substitute $x = 2$ :
$6 (2)^{2} - 5 (2)$
$= 6 (4) - 10$
$= 24 - 10$
$= 14$

Example 49: Simplify $8 p + 3 p - 7$ .

Solution:
$= (8 + 3) p - 7$
$= 11 p - 7$

Example 50: Evaluate $4 x + 9$ when $x = 6$ .

Solution:
Substitute $x = 6$ :
$4 (6) + 9$
$= 24 + 9$
$= 33$

Examples 51 to 60

Example 51: Simplify $5 x + 9 x + 7$ .

Solution:
$= (5 + 9) x + 7$
$= 14 x + 7$

Example 52: Evaluate $7 x - 3$ when $x = 9$ .

Solution:
Substitute $x = 9$ :
$7 (9) - 3$
$= 63 - 3$
$= 60$

Example 53: Simplify $10 y + 6 y - 11$ .

Solution:
$= (10 + 6) y - 11$
$= 16 y - 11$

Example 54: Evaluate $4 x^{2} + 5 x$ when $x = 4$ .

Solution:
Substitute $x = 4$ :
$4 (4)^{2} + 5 (4)$
$= 4 (16) + 20$
$= 64 + 20$
$= 84$

Example 55: Simplify $7 a + 4 - 2 a$ .

Solution:
$= (7 - 2) a + 4$
$= 5 a + 4$

Example 56: Evaluate $10 x + 8$ when $x = 3$ .

Solution:
Substitute $x = 3$ :
$10 (3) + 8$
$= 30 + 8$
$= 38$

Example 57: Simplify $9 y + 3 y + 6$ .

Solution:
$= (9 + 3) y + 6$
$= 12 y + 6$

Example 58: Evaluate $5 x^{2} - 3 x$ when $x = 2$ .

Solution:
Substitute $x = 2$ :
$5 (2)^{2} - 3 (2)$
$= 5 (4) - 6$
$= 20 - 6$
$= 14$

Example 59: Simplify $12 p + 5 - 7 p$ .

Solution:
$= (12 - 7) p + 5$
$= 5 p + 5$

Example 60: Evaluate $6 x + 11$ when $x = 7$ .

Solution:
Substitute $x = 7$ :
$6 (7) + 11$
$= 42 + 11$
$= 53$

Examples 61 to 70

Example 61: Simplify $7 x + 8 x + 12$ .

Solution:
$= (7 + 8) x + 12$
$= 15 x + 12$

Example 62: Evaluate $5 x - 4$ when $x = 6$ .

Solution:
Substitute $x = 6$ :
$5 (6) - 4$
$= 30 - 4$
$= 26$

Example 63: Simplify $8 y + 5 y - 9$ .

Solution:
$= (8 + 5) y - 9$
$= 13 y - 9$

Example 64: Evaluate $7 x^{2} + 2 x$ when $x = 3$ .

Solution:
Substitute $x = 3$ :
$7 (3)^{2} + 2 (3)$
$= 7 (9) + 6$
$= 63 + 6$
$= 69$

Example 65: Simplify $6 a + 7 - 4 a$ .

Solution:
$= (6 - 4) a + 7$
$= 2 a + 7$

Example 66: Evaluate $9 x + 5$ when $x = 2$ .

Solution:
Substitute $x = 2$ :
$9 (2) + 5$
$= 18 + 5$
$= 23$

Example 67: Simplify $10 y - 3 y + 8$ .

Solution:
$= (10 - 3) y + 8$
$= 7 y + 8$

Example 68: Evaluate $6 x^{2} + 3 x$ when $x = 5$ .

Solution:
Substitute $x = 5$ :
$6 (5)^{2} + 3 (5)$
$= 6 (25) + 15$
$= 150 + 15$
$= 165$

Example 69: Simplify $11 p + 6 - 5 p$ .

Solution:
$= (11 - 5) p + 6$
$= 6 p + 6$

Example 70: Evaluate $4 x + 10$ when $x = 4$ .

Solution:
Substitute $x = 4$ :
$4 (4) + 10$
$= 16 + 10$
$= 26$

Examples 71 to 80

Example 71: Simplify $9 x + 2 x + 7$ .

Solution:
$= (9 + 2) x + 7$
$= 11 x + 7$

Example 72: Evaluate $6 x - 5$ when $x = 8$ .

Solution:
Substitute $x = 8$ :
$6 (8) - 5$
$= 48 - 5$
$= 43$

Example 73: Simplify $12 y + 4 y - 6$ .

Solution:
$= (12 + 4) y - 6$
$= 16 y - 6$

Example 74: Evaluate $8 x^{2} + 4 x$ when $x = 3$ .

Solution:
Substitute $x = 3$ :
$8 (3)^{2} + 4 (3)$
$= 8 (9) + 12$
$= 72 + 12$
$= 84$

Example 75: Simplify $7 a + 3 - 2 a$ .

Solution:
$= (7 - 2) a + 3$
$= 5 a + 3$

Example 76: Evaluate $11 x + 9$ when $x = 2$ .

Solution:
Substitute $x = 2$ :
$11 (2) + 9$
$= 22 + 9$
$= 31$

Example 77: Simplify $10 y + 8 y + 4$ .

Solution:
$= (10 + 8) y + 4$
$= 18 y + 4$

Example 78: Evaluate $5 x^{2} - 4 x$ when $x = 4$ .

Solution:
Substitute $x = 4$ :
$5 (4)^{2} - 4 (4)$
$= 5 (16) - 16$
$= 80 - 16$
$= 64$

Example 79: Simplify $8 p + 6 - 5 p$ .

Solution:
$= (8 - 5) p + 6$
$= 3 p + 6$

Example 80: Evaluate $7 x + 11$ when $x = 4$ .

Solution:
Substitute $x = 4$ :
$7 (4) + 11$
$= 28 + 11$
$= 39$

Examples 81 to 90

Example 81: Simplify $6 x + 5 x + 8$ .

Solution:
$= (6 + 5) x + 8$
$= 11 x + 8$

Example 82: Evaluate $8 x - 7$ when $x = 5$ .

Solution:
Substitute $x = 5$ :
$8 (5) - 7$
$= 40 - 7$
$= 33$

Example 83: Simplify $9 y + 4 y - 8$ .

Solution:
$= (9 + 4) y - 8$
$= 13 y - 8$

Example 84: Evaluate $6 x^{2} + 3 x$ when $x = 4$ .

Solution:
Substitute $x = 4$ :
$6 (4)^{2} + 3 (4)$
$= 6 (16) + 12$
$= 96 + 12$
$= 108$

Example 85: Simplify $7 a + 2 - 4 a$ .

Solution:
$= (7 - 4) a + 2$
$= 3 a + 2$

Example 86: Evaluate $9 x + 7$ when $x = 6$ .

Solution:
Substitute $x = 6$ :
$9 (6) + 7$
$= 54 + 7$
$= 61$

Example 87: Simplify $11 y + 5 y - 3$ .

Solution:
$= (11 + 5) y - 3$
$= 16 y - 3$

Example 88: Evaluate $5 x^{2} + 2 x$ when $x = 3$ .

Solution:
Substitute $x = 3$ :
$5 (3)^{2} + 2 (3)$
$= 5 (9) + 6$
$= 45 + 6$
$= 51$

Example 89: Simplify $10 p + 4 - 7 p$ .

Solution:
$= (10 - 7) p + 4$
$= 3 p + 4$

Example 90: Evaluate $8 x + 9$ when $x = 2$ .

Solution:
Substitute $x = 2$ :
$8 (2) + 9$
$= 16 + 9$
$= 25$

Examples 91 to 100

Example 91: Simplify $4 x + 9 x + 3$ .

Solution:
$= (4 + 9) x + 3$
$= 13 x + 3$

Example 92: Evaluate $7 x - 4$ when $x = 5$ .

Solution:
Substitute $x = 5$ :
$7 (5) - 4$
$= 35 - 4$
$= 31$

Example 93: Simplify $6 y + 8 y - 5$ .

Solution:
$= (6 + 8) y - 5$
$= 14 y - 5$

Example 94: Evaluate $9 x^{2} + 2 x$ when $x = 4$ .

Solution:
Substitute $x = 4$ :
$9 (4)^{2} + 2 (4)$
$= 9 (16) + 8$
$= 144 + 8$
$= 152$

Example 95: Simplify $8 a + 7 - 3 a$ .

Solution:
$= (8 - 3) a + 7$
$= 5 a + 7$

Example 96: Evaluate $6 x + 10$ when $x = 5$ .

Solution:
Substitute $x = 5$ :
$6 (5) + 10$
$= 30 + 10$
$= 40$

Example 97: Simplify $7 y + 5 y + 4$ .

Solution:
$= (7 + 5) y + 4$
$= 12 y + 4$

Example 98: Evaluate $4 x^{2} - 5 x$ when $x = 3$ .

Solution:
Substitute $x = 3$ :
$4 (3)^{2} - 5 (3)$
$= 4 (9) - 15$
$= 36 - 15$
$= 21$

Example 99: Simplify $9 p + 6 - 4 p$ .

Solution:
$= (9 - 4) p + 6$
$= 5 p + 6$

Example 100: Evaluate $5 x + 7$ when $x = 6$ .

Solution:
Substitute $x = 6$ :
$5 (6) + 7$
$= 30 + 7$
$= 37$

Variables and Expressions