Algebraic notation, vocabulary and substitution

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Algebraic Notation Basics

In algebra, we often omit the multiplication sign: write 3x instead of 3 × x, and ab instead of a × b.
Coefficients (numbers multiplying letters) are written before the letter: 5y, not y5.
Index notation shows repeated multiplication: x³ means x × x × x, and 2x² means 2 × x × x (not (2x)²).
When a coefficient is a fraction, write it as a fraction: (1/2)x or \frac{1}{2}x. Writing x/2 is also acceptable, but the fraction form is standard.
Division is often shown as a fraction: a/b means a ÷ b.
The identity symbol ≡ means 'is always equal to' for all values of the variable; use it for identities like (x+1)² ≡ x² + 2x + 1.

Vocabulary: Expressions, Equations, and More

An expression is a collection of terms with no equals sign, e.g., 3x + 2.
An equation states that two expressions are equal. Its solutions can include some, all or no values; an identity holds for every permitted value.
A formula is a special equation that shows a relationship between variables, e.g., A = lw (area = length × width).
An identity is true for all values of the variable, e.g., 2(x + 3) ≡ 2x + 6.
An inequality compares expressions using <, >, ≤, ≥, e.g., 3x + 2 > 11.
A term is a single number, variable, or product of numbers and variables, e.g., in 3x + 2, the terms are 3x and 2.
A factor is a number or expression that divides exactly into a term, e.g., in 6x, the factors include 2, 3, 6, x, 2x, 3x, 6x.

Substitution Basics

Substitution means replacing a variable with a given numerical value.
Always use brackets when substituting negative numbers: substitute x = -3 into 2x² as 2(-3)², not 2-3².
Follow the order of operations (BIDMAS/BODMAS): brackets, indices, division/multiplication, addition/subtraction.
When substituting into a formula, write the formula first, then replace letters with numbers.
For example, substitute a = 2, b = 5 into a + 3b: 2 + 3 × 5 = 2 + 15 = 17.
When substituting fractions, treat them as numbers: if x = 1/2, then 4x = 4 × (1/2) = 2.

Substituting Negative Numbers

Watch the sign when substituting negatives: (-4)² = 16, not -16.
For a term like -x² with x = 3, the negative applies after squaring: -3² = -9, but (-3)² = 9.
Use brackets to avoid sign errors: substitute x = -2 into 3x - 5 as 3(-2) - 5 = -6 - 5 = -11.
When substituting into a squared term, square the whole value: if x = -3, then x² = (-3)² = 9.
If the expression has a coefficient, square only the variable: 2x² with x = -3 gives 2 × (-3)² = 2 × 9 = 18, not (2 × -3)².

Substituting into Scientific and Geometric Formulas

You may be asked to substitute into formulas from science, e.g., speed = distance/time (v = s/t).
For example, if distance = 100 m and time = 20 s, then v = 100/20 = 5 m/s.
Geometric formulas include area of a rectangle A = lw, area of a triangle A = 1/2 × b × h, and volume of a cuboid V = lwh.
Substitute the given values and calculate step by step, showing your working.
If a formula uses a fraction, like P = \frac{F}{A}, substitute F and A directly: if F = 20 and A = 5, then P = 20/5 = 4.
Always include units in your final answer when the question gives units.

Common Misconceptions and Errors

3x does not mean 3 + x; it means 3 × x.
2x and x² are different: 2x = x + x, while x² = x × x.
When squaring a product like (3x)², square both the coefficient and the variable: (3x)² = 9x², not 3x².
Don't square only part of a term: (2x)² = 4x², not 2x².
Expressions can be evaluated or simplified. Equations and inequalities can be solved to find allowed variable values.
When substituting negatives, remember that (-4)² = 16, and -4² = -16 (the negative is not squared unless in brackets).

Using the Identity Symbol

The identity symbol ≡ shows that two expressions are always equal for every value of the variable.
For example, 3(x + 2) ≡ 3x + 6 because the distributive law always holds.
Use ≡ only for identities, not for equations that are true for specific values.
To check if a statement is an identity, expand or simplify both sides and see if they match exactly.
An identity must be true for every permitted value. Check by expanding or simplifying; a symbol alone does not prove a claim is true.

Writing Expressions from Descriptions

Translate word problems into algebraic expressions using letters for unknown quantities.
For example, 'three more than a number' becomes x + 3.
'The product of a number and 5' becomes 5x.
'One quarter of a number' becomes x/4 or \frac{x}{4}.
'The sum of a number and its square' becomes x + x².
Use brackets when needed: 'twice the sum of a and b' becomes 2(a + b).

Evaluating Expressions with Fractions

Substitute fractional values carefully, using brackets: if x = 1/2, then 3x = 3 × (1/2) = 3/2.
When the expression has a fraction, like x/2, substitute and simplify: if x = 3, then 3/2.
For expressions like 2/x, substitute x = 1/4: 2 ÷ (1/4) = 2 × 4 = 8.
Remember that dividing by a fraction is the same as multiplying by its reciprocal.
Simplify your final answer to lowest terms.

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1.In algebra, what does the expression 5y mean?
Easy
- A5 plus y
- B5 times y
- C5 divided by y
- Dy to the power 5
2.The expression 2x and x² are equivalent.
Easy
True or false?
3.Which expressions use standard compact algebraic notation, with multiplication signs omitted? Select all that apply.
Easy
- A3x
- B2a + b
- Cx3
- Da × b
- Ep²
4.Which of the following is an identity?
Medium
- Ax + 2 = 5
- B3(x + 2) = 3x + 6
- Cy = 2x + 1
- Dx² = 4
5.Given that p = -2 and q = 3, what is the value of p² + 2q?
Medium
- A10
- B2
- C-2
- D4
6.Match each algebraic expression with its meaning.
Medium
- 2x
- x²
- x/2
- x multiplied by x
- x divided by 2
- 2 multiplied by x
7.Put the following steps in the correct order to evaluate the expression 2x² - 3 when x = -2.
Medium
- Calculate x²: (-2)² = 4
- Multiply by 2: 2 × 4 = 8
- Subtract 3: 8 - 3 = 5
8.The formula for converting temperature from Celsius to Fahrenheit is F = \frac{9}{5}C + 32. What is the temperature in Fahrenheit when C = 20?
Hard
- A68°F
- B52°F
- C72°F
- D100°F

Plus de questions

9.Which of the following are equivalent to x² - 4? (Select all that apply)
Hard
- A(x - 2)(x + 2)
- Bx² - 2²
- Cx² - 4x + 4
- D(x + 2)(x - 2)
- Ex² + 4
10.If x = \frac{1}{2}, then x² = \frac{1}{4}.
Hard
True or false?
11.Which of the following is the correct expansion of 3(x + 4) - 2(x - 1)?
Challenge
- Ax + 10
- Bx + 14
- C5x + 10
- D5x + 14
12.For x ≠ 0, match each expression with its simplified form.
Challenge
- 2x + 3x
- x² × x³
- x⁵ ÷ x²
- 5x
- x⁵
- x³
13.Match each algebraic term with its correct meaning.
Easy
- 3x
- x³
- 3 + x
- 3x²
- three times x
- x cubed
- three plus x
- three times x squared
14.Put the following steps in the correct order to evaluate the expression 5 - 2x² when x = -3.
Medium
- Substitute x = -3: 5 - 2(-3)²
- Calculate the power: (-3)² = 9
- Multiply: 2 × 9 = 18
- Subtract: 5 - 18 = -13
15.In the expression 7x², the coefficient of x² is 7.
Easy
True or false?
16.The formula for the area of a trapezium is A = \frac{1}{2}(a + b)h. Find the value of A when a = 5, b = 7, and h = 4.
Medium
- A24
- B48
- C12
- D34
17.When x = -3, what is the value of the expression 2x² - x?
Hard
- A21
- B15
- C-15
- D-21
18.Which of the following is the correct algebraic notation for 'the product of 3 and x'?
Easy
- A3x
- Bx3
- C3 + x
- D3/x
19.What is the value of the expression 4a - 2b when a = 3 and b = -5?
Medium
- A22
- B2
- C-2
- D12
20.The expression 5y means '5 plus y'.
Easy
True or false?
21.Given that p = -3 and q = 2, what is the value of p² - 2q?
Medium
- A5
- B13
- C-13
- D-5
22.Put the following steps in the correct order to evaluate the expression 2x² - 3x when x = -2.
Medium
- Substitute x = -2 into the expression.
- Calculate 2x² = 2 \times 4 = 8.
- Calculate -3x = -3 \times -2 = 6.
- Add the results: 8 + 6 = 14.
23.The formula for the volume of a cylinder is V = \π r² h. Using \π \approx 3.14, what is the volume, in cm³, of a cylinder with radius 2 cm and height 5 cm?
Medium
- A62.8
- B31.4
- C20
- D125.6
24.The area of a triangle is given by A = \frac{1}{2} b h. What is the area, in cm², of a triangle with base 6 cm and height 8 cm?
Medium
- A24
- B48
- C14
- D96
25.The formula for converting temperatures is F = \frac{9}{5}C + 32. What is the temperature in Fahrenheit when C = -10?
Medium
- A14
- B-14
- C50
- D-50
26.The formula for the perimeter of a rectangle is P = 2(l + w). Which of the following are equivalent to the perimeter when l = 4 and w = 3? (Select all that apply)
Hard
- A2(4 + 3)
- B2 × 4 + 2 × 3
- C14
- D2 × 7
- E4 × 3
27.When substituting x = -2 into the expression 3x², the value is 12.
Medium
True or false?
28.In algebra, the expression 6p means 6 multiplied by p.
Easy
True or false?
29.Which of the following is the correct algebraic notation for 'the sum of x and 4, then multiplied by 3'?
Easy
- A3(x + 4)
- B3x + 4
- Cx + 4 × 3
- D3 + x + 4
30.What is the value of the expression 5 - 2x² when x = -3?
Easy
- A-13
- B23
- C-31
- D41
31.Put the following steps in the correct order to evaluate the expression 3x² - 2x when x = -1.
Medium
- Substitute x = -1: 3(-1)² - 2(-1)
- Calculate the power: 3(1) - 2(-1)
- Multiply: 3 + 2
- Add: 5
32.Given that x = \frac{1}{2}, what is the value of 6x² - 3x?
Hard
- A0
- B1.5
- C-1.5
- D0.75
33.Match each algebraic expression to its correct meaning.
Easy
- 3x
- x³
- 3/x
- x + 3
- 3 divided by x
- 3 times x
- x cubed
- 3 more than x
34.Put the following steps in the correct order to evaluate the expression 2x² - 3x when x = -2.
Medium
- Substitute x = -2: 2(-2)² - 3(-2)
- Evaluate the exponent: 2(4) - 3(-2)
- Multiply: 8 + 6
- Add: 14
35.Which of the following are true statements when substituting x = -3 into the expression x² - 2x? (Select all that apply)
Medium
- AThe value is 15.
- BThe value is 3.
- CThe value is (-3)² - 2(-3).
- DThe value is 9 + 6.
- EThe value is -3.
36.The expression 4x means 4 plus x.
Easy
True or false?
37.What is the value of 3a² when a = -2?
Medium
- A-12
- B12
- C-6
- D6
38.(3x)² is equivalent to 3x² for all values of x.
Medium
True or false?
39.Which of the following is the correct algebraic notation for 'the product of 3 and x, divided by 2'?
Medium
- A\frac{3x}{2}
- B\frac{3}{2x}
- C\frac{3+x}{2}
- D3x+2
40.Which of the following are equivalent to x² + 4x? (Select all that apply)
Medium
- Ax(x + 4)
- Bx² + 4x
- C(x + 2)²
- Dx(x) + 4x
41.Fill in the blanks to complete the algebraic vocabulary definitions.
Easy
A mathematical statement with an equals sign that is true for certain values is called an ____. A statement that is true for all values is called an ____. A statement with an equals sign that can be used to calculate something is called a ____. A statement with a greater than or less than sign is called an ____.
42.Given that a = 3 and b = 4, work out the value of 2a + b.
Easy
43.Substitute x = -3 into the expression x² - 2x and find the value.
Medium
44.Use x for the number. Write an expression for the sum of the number and 5, then multiplied by 3.
Medium
45.Fill in the blank: The value of 5x² when x = -1 is ___.
Medium
The value of 5x² when x = -1 is ____.
46.Find the area of a trapezium with a = 5 cm, b = 7 cm, and h = 4 cm.
Medium
47.A trapezium has parallel sides a = 3 cm and b = 9 cm, and perpendicular height h = 5 cm. Using A = ½(a + b)h, calculate its area.
Medium
48.Write down the coefficient of x in the expression 5x² - 3x + 2.
Hard
49.Given that y = 3x² - 2x + 1, find the value of y when x = -2.
Challenge
50.Fill in the blank: If a = 2 and b = -3, then a² - b² = ___.
Challenge
If a = 2 and b = -3, then a² - b² = ____.
51.Write down the term in the expression 4x² - 3x + 5 that is a constant.
Easy
52.Substitute x = 3 into the expression 4x² and find the value.
Medium
53.Explain why the statement 3x + 2 = 5 is an equation, not an identity.
Medium
54.Explain why the statement 3x² is not equivalent to (3x)².
Hard
55.Explain why (3x)² is not equivalent to 3x².
Hard
56.A student writes: 'The expression 4x - 3 has a value that changes when x changes.' Is this statement correct? Explain your answer.
Hard
57.Explain why the statement 3x + 2 = 5 is an equation, not an expression.
Medium
58.Given that a = 3 and b = 4, work out the value of 2a + b.
Easy
- A10
- B12
- C9
- D11
59.Substitute x = -3 into the expression x² - 2x and find the value.
Medium
- A15
- B14
- C17
- D16
60.Find the area of a trapezium with a = 5 cm, b = 7 cm, and h = 4 cm.
Medium
- A26 cm²
- B24 cm²
- C22 cm²
- D28 cm²
61.A trapezium has parallel sides a = 3 cm and b = 9 cm, and perpendicular height h = 5 cm. Using A = ½(a + b)h, calculate its area.
Medium
- A27 cm²
- B33 cm²
- C30 cm²
- D36 cm²
62.Given that y = 3x² - 2x + 1, find the value of y when x = -2.
Challenge
- A18
- B16
- C17
- D19
63.Substitute x = 3 into the expression 4x² and find the value.
Medium
- A36
- B40
- C44
- D32

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