Introduction To Algebra

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Algebraic Notation

Algebra uses letters (variables) to represent general or unknown numbers.
Algebraic notation writes calculations using letters: e.g., `a + b`, `ab` means `a × b`, `a/b` means `a ÷ b`.
Multiplication is implied: `3ab` means `3 × a × b`.
Order of operations applies: brackets, powers, then multiplication/division, then addition/subtraction.
Powers and roots are written as with numbers: `a²` means `a × a`, `√a` means square root of `a`.
Brackets work as with numbers: `3(a + b)` means `3 × (a + b)`.

Algebraic Vocabulary

A term is a variable, a number (constant), or a product of numbers and variables (e.g., `5x`, `4xy`).
The number in front of a variable is the coefficient (e.g., coefficient of `x` in `6x` is 6).
A factor divides a term exactly; e.g., factors of `3x` are 1, 3, x, 3x.
An expression has no equals sign (e.g., `2x + 5y`).
An equation has an equals sign and can be solved (e.g., `2x = 10`).
A formula is a rule linking quantities (e.g., `w = mg`); substituting values turns it into an equation.

Substitution

Substitution replaces letters in a formula with given numbers.
Always use brackets around negative numbers when substituting (e.g., `x = -3` → `x² = (-3)² = 9`).
Follow order of operations: brackets, powers, multiplication/division, addition/subtraction.
Substitution can produce an equation to solve for an unknown variable.
Example: If `P = 2l + 2w`, `P = 20`, `w = 4`, then `20 = 2l + 8` → `l = 6`.

Collecting Like Terms

Like terms have exactly the same variables and powers (e.g., `2x` and `3x`; `5xy` and `-7xy`).
Unlike terms have different variables or powers (e.g., `2x` and `3y`; `4x²` and `6x`).
To collect like terms, add or subtract their coefficients.
Keep the sign with its term; `2x - 3y` is the same as `2x + (-3y)`.
Simplify: `8a - 5b - 6a + 4b` → `(8a - 6a) + (-5b + 4b)` → `2a - b`.
Do not leave `1x` or `-1x`; write `x` or `-x`.

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1.Which of the following is an expression?
Easy
- A2x + 5 = 4
- B7x - 9
- Cx = vt - w
- D3 + 4 = 7
2.Which of the following is an equation?
Easy
- A2x + 5
- B7x - 9
- C2x + 5 = 4
- Dx = vt - w
3.Simplify: 6a - 3b + 2a - 4b
Medium
- A8a - 7b
- B8a + 7b
- C4a - 7b
- D8a - b
4.Simplify: 4x + 3y + 2x - 8y
Medium
- A6x - 5y
- B6x + 5y
- C2x - 5y
- D6x - 11y
5.W = 3a + 5c. Find the value of W when a = 6 and c = 2.
Easy
- A28
- B23
- C48
- D36
6.Simplify: 3c - 5d - c + 2d
Medium
- A2c - 3d
- B2c - 7d
- C4c - 3d
- D2c + 3d
7.Simplify: 5w + 3h - 7w + 8h
Medium
- A-2w + 11h
- B2w + 11h
- C-2w - 5h
- D12w + 11h
8.Simplify: 5c - d - 3d - 2c
Medium
- A3c - 4d
- B3c + 4d
- C7c - 4d
- D3c - 2d

Mehr Fragen

9.Simplify: 5t + 4t - 2t
Easy
- A7t
- B11t
- C3t
- Dt
10.Simplify: 8a + 3b - 2a + b
Medium
- A6a + 4b
- B10a + 4b
- C6a + 2b
- D6a - 4b
11.A = 1/4 b c². Calculate the value of A when b = 3 and c = 6.
Medium
- A27
- B9
- C54
- D12
12.T = 3a² / b. Find the value of T when a = 4 and b = 5.
Medium
- A9.6
- B48
- C12
- D4.8
13.Kyra has a faulty thermometer. It always shows the temperature as 2°C higher than the actual temperature. The temperature on the thermometer is T °C. Write an expression, in terms of T, for the actual temperature.
Medium
- AT - 2
- BT + 2
- C2 - T
- DT/2
14.Calculate the value of 4x² + xy when x = 3 and y = -2.
Medium
- A30
- B36
- C18
- D24
15.Simplify: 4x - 12y + 10x + 25y
Medium
- A14x + 13y
- B14x - 13y
- C6x + 13y
- D14x + 37y
16.Simplify: 2b + 3b - a - b + 3a
Medium
- A4b + 2a
- B4b - 2a
- C6b + 2a
- D4b + 4a
17.Simplify: 7x + 2y - 3x + 5y
Medium
- A4x + 7y
- B10x + 7y
- C4x - 3y
- D10x - 3y
18.Simplify: 9p - 4q - 5p + 7q
Medium
- A4p + 3q
- B14p + 3q
- C4p - 11q
- D14p - 11q
19.Simplify: 6m + 3n - 2m - 8n
Medium
- A4m - 5n
- B8m - 5n
- C4m + 11n
- D8m + 11n
20.Simplify: 4a + 7b - a - 2b
Medium
- A3a + 5b
- B5a + 5b
- C3a + 9b
- D5a + 9b
21.Simplify: 2x + 5y + 3x - y
Medium
- A5x + 4y
- B5x + 6y
- Cx + 4y
- Dx + 6y
22.Simplify: 3c + 2d + 5c - 7d
Medium
- A8c - 5d
- B8c + 9d
- C2c - 5d
- D2c + 9d
23.Simplify: 8t - 3u - 5t + 6u
Medium
- A3t + 3u
- B13t + 3u
- C3t - 9u
- D13t - 9u
24.Simplify: 6x + 2y - 4x - 3y
Medium
- A2x - y
- B10x - y
- C2x + 5y
- D10x + 5y
25.Given that a = 5 and b = -2, find the value of 3a + 4b.
Easy
- A7
- B23
- C15
- D11
26.Given that x = 3 and y = -4, find the value of 2x² - 3y.
Medium
- A30
- B6
- C18
- D42
27.The formula for the area of a trapezium is A = (a + b)h/2. Find the value of A when a = 4, b = 6, and h = 5.
Easy
- A25
- B50
- C15
- D30
28.Simplify: 3(2x + 4) - 2(3x - 1)
Medium
- A14
- B12x + 14
- C12x + 10
- D10
29.If P = 2l + 2w, and P = 30, w = 5, find l.
Medium
- A10
- B12.5
- C20
- D5
30.Given that c = 5a - 2b, find the value of c when a = 8 and b = -3.
Medium
- A46
- B34
- C40
- D28
31.A pen costs 60 cents and a ruler costs 29 cents. Write an expression for the total cost, in cents, of x pens and y rulers.
Easy
- A60x + 29y
- B60y + 29x
- C89(x + y)
- D60x + 29
32.Simplify: 9x - 2y - 5x - y
Easy
- A4x - 3y
- B4x - y
- C14x - 3y
- D14x - y
33.A rectangle has length l cm and width w cm. Its perimeter is given by P = 2l + 2w. If the perimeter is 30 cm and the width is 6 cm, what is the length?
Medium
- A9 cm
- B12 cm
- C15 cm
- D18 cm
34.The formula for the area of a trapezium is A = (a + b)h / 2. Work out the value of h when A = 38.64, a = 5.5, and b = 3.7.
Medium
- A8.4
- B4.2
- C16.8
- D2.1
35.c = 5a - 2b. Make a the subject of the formula.
Medium
- Aa = (c + 2b) / 5
- Ba = (c - 2b) / 5
- Ca = 5c + 2b
- Da = (2b - c) / 5
36.The angles of a triangle are x, 2x, and 3x. Find the value of x.
Medium
- A30°
- B60°
- C90°
- D20°
37.A triangle has sides of length (x+2) cm, (2x-1) cm, and (3x-1) cm. Its perimeter is 30 cm. Find x.
Medium
- A4
- B5
- C6
- D7
38.What is the coefficient of x in the term 7x?
Easy
- A7
- Bx
- C7x
- D1
39.Which of the following is a term?
Easy
- A3x + 2
- B5y
- C2 = 2
- Dx + y = 3
40.Write an expression for 'a number n multiplied by 5, then increased by 3'.
Easy
- A5n + 3
- B5(n + 3)
- C5n - 3
- Dn + 15
41.What is the value of 2x + 3 when x = 4?
Easy
- A11
- B14
- C9
- D8
42.Simplify: 3a + 2b - a + 5b
Easy
- A2a + 7b
- B4a + 7b
- C2a + 3b
- D3a + 7b
43.In algebra, the expression 'ab' means a + b.
Easy
True or false?
44.What is the coefficient of y in the term -5y?
Easy
- A-5
- B5
- Cy
- D-y
45.The expression 4x² means 4 × x × x.
Easy
True or false?
46.Simplify: 4x - 3y + 2x + 5y
Easy
- A6x + 2y
- B6x - 8y
- C2x + 2y
- D4x + 2y
47.The expression 3ab means 3 + a + b.
Easy
True or false?
48.Match each algebraic term to its coefficient of x.
Medium
- 5x
- -3x²
- x
- 0x
- 5
- -3
- 1
- 0
49.Arrange these steps in the correct order to simplify 3x + 5 - 2x + 3.
Medium
- 3x - 2x + 5 + 3
- x + 8
- 3x + 5 - 2x + 3
- (3x - 2x) + (5 + 3)
50.Select all the expressions that are equivalent to 6x + 4y - 2x - 3y.
Medium
- A4x + y
- B8x + 7y
- C4x - y
- D6x - 2x + 4y - 3y
- E2(2x + y)
51.Given that y = 3x - 5, find the value of y when x = 4.
Easy
52.Complete the sentence.
Easy
In algebra, the expression a/b means a ____ b.
53.Find the value of 5p + 2q when p = 3 and q = -1.
Medium
54.Given that a = 3 and b = -2, find the value of 2a² - 3b.
Medium
55.From the list: 3x + 2 = 8, 5y - 1, 4z, write down an equation.
Easy
56.Given that y = 3x - 5, find the value of y when x = 4.
Easy
- A8
- B7
- C9
- D6
57.Find the value of 5p + 2q when p = 3 and q = -1.
Medium
- A13
- B14
- C15
- D12

Lernnotizen

## Introduction to Algebra - Quick Revision ### Algebraic Notation - Letters represent unknown numbers (variables). - Multiplication: no symbol, e.g., `ab` means a × b. - Division: use fractions, e.g., `a/b` means a ÷ b. - Powers: `a²` means a × a. - Brackets: `3(a+b)` means 3 × (a+b). ### Algebraic Vocabulary - **Term**: a variable, constant, or product (e.g., `x`, `20`, `5x`). - **Coefficient**: the number in front of a variable (e.g., in `6x`, coefficient is 6). - **Expression**: no equals sign (e.g., `2x+5y`). - **Equation**: has an equals sign, can be solved (e.g., `2x=10`). - **Formula**: a rule linking quantities (e.g., `P=2l+2w`). ### Substitution - Replace letters with given numbers. - Use brackets around negative numbers: `x²` when x = -4 → `(-4)² = 16`. - Follow order of operations (BIDMAS). - Example: If `A = ¼ bc²`, b=3, c=6 → `A = ¼ × 3 × 36 = 27`. ### Collecting Like Terms - Like terms have same variable(s) and same powers. - Add/subtract coefficients only. - Example: `8a - 5b - 6a + 4b = (8a-6a) + (-5b+4b) = 2a - b`. - Simplify `1x` to `x`, `-1x` to `-x`. ### Key Formulas - Perimeter of rectangle: `P = 2l + 2w`. - Area of trapezium: `A = (a+b)h/2`. - From past paper: `c = 5a - 2b`; making a subject: `a = (c+2b)/5`.

Variable: A letter representing an unknown number.
Coefficient: The number multiplying a variable (e.g., in 5x, coefficient is 5).
Term: A single number, variable, or product of numbers and variables.
Expression: An algebraic statement without an equals sign.
Equation: An algebraic statement with an equals sign that can be solved.
Formula: A rule or relationship written in algebraic shorthand.
Substitution: Replacing a variable with a given numerical value.
Like terms: Terms with exactly the same variables and powers.
Collecting like terms: Simplifying an expression by adding/subtracting coefficients of like terms.

Karteikarten

What is algebraic notation?

Algebraic notation uses letters to represent numbers. For example, 'ab' means a × b, and 'a/b' means a ÷ b.

How do you write multiplication in algebra?

No symbol is used; e.g., 'ab' means a × b, and '3ab' means 3 × a × b.

How do you write division in algebra?

Use fractions: 'a/b' means a ÷ b.

What does 'a²' mean?

a² means a × a (a squared).

What does '4a²' mean?

4a² means 4 × a × a (4 times a squared).

What does '√a' mean?

√a means the square root of a.

What does '3(a + b)' mean?

It means 3 × (a + b): add a and b first, then multiply by 3.

What is a term in algebra?

A term is a number, a variable, or a product of numbers and variables, e.g., 5x, 20, or x².

What is a coefficient?

The number in front of a variable; e.g., in 6x, the coefficient is 6; in -5y, it is -5.

What is a constant?

A constant is a term that does not contain a variable, e.g., 20.

What is an expression?

An algebraic statement without an equals sign, e.g., 2x + 5y.

What is an equation?

An algebraic statement with an equals sign that can be solved, e.g., 2x = 10.

What is a formula?

A rule or relationship written in letters, e.g., w = mg. It becomes an equation when values are substituted.

What does substitution mean?

Replacing a variable with a given number. Use brackets for negative numbers, e.g., if x = -4, then x² = (-4)² = 16.

Simplify: 6a - 3b + 2a - 4b

8a - 7b

Simplify: 4x + 3y + 2x - 8y

6x - 5y

Simplify: 3c - 5d - c + 2d

2c - 3d

Simplify: 5w + 3h - 7w + 8h

-2w + 11h

Simplify: 5c - d - 3d - 2c

3c - 4d

Simplify: 5t + 4t - 2t

Simplify: 8a + 3b - 2a + b

6a + 4b

Simplify: 4x - 12y + 10x + 25y

14x + 13y

Simplify: 2b + 3b - a - b + 3a

4b + 2a

Simplify: 9x - 2y - 5x - y

4x - 3y

If W = 3a + 5c, find W when a = 6 and c = 2.

W = 3(6) + 5(2) = 18 + 10 = 28

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