Algebra Toolkit

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レッスンノート

Algebraic Notation

Algebra uses letters to represent general or unknown numbers; letters are called variables.
Multiplication is written without a symbol: ab means a × b, 3ab means 3 × a × b.
Division is written as a fraction: a/b means a ÷ b.
Powers (indices) and roots work as with numbers: a² means a × a, √a means square root of a.
Brackets indicate the operation inside is done first: 3(a + b) means 3 × (a + b).
Order of operations (BIDMAS) applies to algebra just as with numbers.

Algebraic Vocabulary

A term is a variable, a number (constant), or a product of numbers and variables (e.g., 5x, 4xy²).
The coefficient is the number in front of a variable: in -5y, the coefficient is -5.
A factor divides a term exactly: factors of 3x are 1, 3, x, and 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 relating quantities (e.g., P = 2l + 2w); substituting values turns it into an equation.

Substitution

Substitution means replacing a variable with a given number.
Always use brackets around negative numbers when substituting: if x = -4, then x² = (-4)² = 16.
Follow the order of operations (BIDMAS) carefully after substitution.
Substitution can produce an equation to solve (e.g., 20 = 2l + 8 gives l = 6).
On a calculator, put brackets around any substituted negative numbers.

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).
Collect like terms by adding or subtracting their coefficients: 2x + 3x = 5x; 4y - 10y = -6y.
The sign in front of a term belongs to that term; terms can be reordered with their signs.
Simplify coefficients of 1 or -1 to just the variable (e.g., 1x → x, -1y → -y).
Collect different types of like terms separately: 2x + 4y + 5x - 3y = 7x + y.

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練習問題

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1.Simplify 3a + 7b - 4a + b.
Easy
- A-a + 8b
- B7a + 8b
- C-a + 6b
- D7a + 6b
2.Find the value of 7x + 3y when x = 12 and y = -6.
Easy
- A66
- B102
- C30
- D-18
3.Simplify 5c - d - 3d - 2c.
Easy
- A3c - 4d
- B7c - 4d
- C3c - 2d
- D7c - 2d
4.Complete the statement: When w = ______, 10w = 70.
Easy
- A7
- B700
- C0.7
- D60
5.Use the formula s = ut + 1/2 at². Calculate s when u = 5, t = 10 and a = 3.
Medium
- A200
- B80
- C350
- D65
6.Simplify 2p - q - 3q - 5p.
Easy
- A-3p - 4q
- B3p - 4q
- C-3p + 2q
- D7p - 4q
7.The body mass index B is given by B = m / h². Usman has mass 50 kg and height 1.57 m. Work out B correct to one decimal place.
Medium
- A20.3
- B31.8
- C20.2
- D32.0
8.Find the value of y when m = -2, x = -7 and c = -3 in y = mx + c.
Easy
- A11
- B-17
- C-11
- D17

もっと問題

9.A = 4bc. If A = 100 and b = 2, work out the value of c.
Easy
- A12.5
- B50
- C200
- D25
10.x = 0.7. Work out the value of (x+1)² / (2x). Write down all the figures on your calculator display.
Medium
- A2.064285714
- B2.0643
- C2.06
- D2.064
11.w = 5y² - y³. Work out w when y = -2.
Medium
- A28
- B12
- C-12
- D36
12.y = x⁴ - 4x³. Find y when x = -1.
Medium
- A5
- B-3
- C-5
- D3
13.One solution of ax² + a = 150 is x = 7. Find the value of a.
Medium
- A3
- B25
- C6
- D150
14.y = p - 2qx². p = -10, q = 3, x = -5. Work out y.
Medium
- A-160
- B140
- C-140
- D160
15.Simplify 7a + 2b - 3a + 8a + 6b + 2a - b.
Easy
- A14a + 7b
- B14a + 9b
- C20a + 7b
- D20a + 9b
16.Kelly is trying to find values of w for which 3w - w³ = 2. Her values are 1 and -1. Are her values correct?
Medium
- ABoth correct
- BOnly 1 correct
- COnly -1 correct
- DNeither correct
17.What is the coefficient of x in the term -7x?
Easy
- A-7
- B7
- Cx
- D-1
18.Which of the following is an expression?
Easy
- A3x + 2
- B3x + 2 = 5
- Cy = 3x + 2
- D3x + 2 > 5
19.Simplify 4x² + 3x - 2x² + 5x.
Easy
- A2x² + 8x
- B6x² + 8x
- C2x² + 2x
- D6x² + 2x
20.If a = 3 and b = -2, find the value of 2a² - 3b.
Medium
- A24
- B6
- C18
- D12
21.Solve for x: 2(x + 3) = 5x - 9.
Medium
- A5
- B3
- C4
- D2
22.The formula for the area of a trapezium is A = 1/2 (a + b)h. Find h when A = 30, a = 4, b = 6.
Medium
- A6
- B3
- C10
- D2
23.Which of the following is a formula?
Easy
- Av = u + at
- B3x + 2
- C2x = 10
- Dx > 5
24.What is the value of 10 - 2x when x = 3?
Easy
- A4
- B16
- C24
- D-16
25.Simplify 3a − 5b − a + 2b.
Easy
- A2a − 3b
- B4a − 7b
- C2a − 7b
- D4a − 3b
26.f = 3g + 7h. Work out f when g = −5 and h = 2.
Easy
- A−1
- B29
- C−29
- D1
27.y = mx + c. Find y when m = −3, x = −2, and c = −8.
Easy
- A−2
- B2
- C14
- D−14
28.G = (c² − 4)/c. Find G when c = −5.
Medium
- A−4.2
- B4.2
- C−21/5
- D21/5
29.Amy has x beads. Billy has three more beads than Amy. Carly has four times as many beads as Billy. Which expression describes the number of beads that Carly has?
Medium
- A4x + 3
- B3x + 4
- C4(x + 3)
- Dx + 12
30.Which of the following is an algebraic expression?
Easy
- A3x + 2 = 5
- B4y - 7
- C5 = 2 + 3
- Dx > 2
31.What is the coefficient of x in the term -3x?
Easy
- A3
- B-3
- Cx
- D-x
32.The term '5' is a constant.
Easy
True or false?
33.If a = -3 and b = 4, what is the value of 2a + b²?
Medium
- A10
- B22
- C-2
- D14
34.Simplify 5x + 3y - 2x + y.
Easy
- A3x + 4y
- B7x + 4y
- C3x + 2y
- D7x + 2y
35.The terms 4x and 4x² are like terms.
Easy
True or false?
36.Which of the following are like terms with 3xy? (Select all that apply)
Medium
- A2xy
- B3x
- C5yx
- D3x²y
- Exy
37.Match each algebraic term with its coefficient of x.
Medium
- 3x
- -5x
- x
- 0x
- 3
- -5
- 1
- 0
38.Arrange the following steps to simplify 5a + 3b - 2a + 4b in the correct order.
Medium
- Combine a terms: 5a - 2a = 3a
- Combine b terms: 3b + 4b = 7b
- Write the simplified expression: 3a + 7b
- Identify like terms
39.Which of the following are like terms with 4xy? (Select all that apply)
Medium
- A3xy
- B4x²y
- C-2xy
- D4y
- Eyx
40.Complete the sentence using the correct algebraic vocabulary.
Easy
In the expression 7x + 3, the number 7 is called the ____ of x.
41.Write down an algebraic expression for 'three times a number added to five'.
Easy
42.Substitute x = 3 and y = -2 into the expression 2x² - 3y. What is the value?
Medium
43.Use the formula v = u + at. Find v when u = 10, a = -2, and t = 4.
Easy
44.Substitute x = -3 and y = 2 into the expression 2x² - 3y and find the value.
Medium
45.Explain the difference between an expression and an equation. Give an example of each.
Medium
46.Substitute x = 3 and y = -2 into the expression 2x² - 3y. What is the value?
Medium
- A24
- B28
- C22
- D26
47.Use the formula v = u + at. Find v when u = 10, a = -2, and t = 4.
Easy
- A3
- B4
- C1
- D2
48.Substitute x = -3 and y = 2 into the expression 2x² - 3y and find the value.
Medium
- A12
- B11
- C13
- D14

復習ノート

## Algebra Toolkit - Night Before Exam Summary ### 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 number, variable, or product (e.g., `5x`, `-3y²`). - **Coefficient**: the number in front of a variable (e.g., in `6x`, coefficient is 6). - **Expression**: no equals sign (e.g., `2x + 5`). - **Equation**: has an equals sign, can be solved (e.g., `2x = 10`). - **Formula**: a rule relating quantities (e.g., `s = ut + ½at²`). ### Substitution - Replace letters with given numbers. - Use brackets around negative numbers: `(-3)`. - Follow order of operations (BIDMAS). - Example: If `x = 2`, `y = -4`, then `2x(x + 3y) = 2×2×(2 + 3×(-4)) = -40`. ### Collecting Like Terms - Like terms have same variables and powers (e.g., `2x` and `5x`). - Add/subtract coefficients only. - Example: `8a - 5b - 6a + 4b = 2a - b`. ### Key Formulas - `s = ut + ½at²` (displacement with constant acceleration). - `y = mx + c` (straight line). - `B = m/h²` (body mass index). ### Tips - Simplify `1x` to `x`. - For negative substitution, e.g., `x = -4`, `x² = (-4)² = 16`.

Variable: A letter representing an unknown or general number.
Coefficient: The number multiplying a variable (e.g., in 5x, coefficient is 5).
Term: A single number, variable, or product (e.g., 3x, -2y²).
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 between quantities using letters.
Substitution: Replacing a variable with a given number.
Like terms: Terms with exactly the same variables and powers.
Collecting like terms: Simplifying by adding/subtracting coefficients of like terms.

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