Algebraic expressions, equations and formulae

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What is an algebraic expression?

An algebraic expression is a maths sentence with numbers, letters, and signs like +, −, ×, ÷.
Example: 3n + 5 is an algebraic expression.
The variable is a letter that stands for an unknown number. In 3n + 5, n is the variable.
The constant is a number on its own. In 3n + 5, the constant is 5.
The coefficient is the number multiplying the variable. In 3n, the coefficient is 3.
A term is a part of an expression separated by + or −. In 3n + 5, the terms are 3n and 5.

Writing expressions

Use a letter for an unknown number. For example, 'a number plus 3' can be written as x + 3.
Multiplication is often shown by writing the number and letter together, like 4y means 4 × y.
Division can be written as a fraction: a/2 means a ÷ 2.
When you write expressions, put the number (coefficient) before the letter: 5x, not x5.
If the coefficient is 1, you can just write the letter: 1x is written as x.

Evaluating expressions

To evaluate an expression, replace the variable with a number and do the maths.
For 3n + 5, if n = 2, then 3×2 + 5 = 6 + 5 = 11.
Always follow the order of operations: brackets first, then × and ÷, then + and −.
Check your work by doing the steps again or using a different method.

Equivalent expressions

Equivalent expressions have the same value for any number you put in.
For example, 2x + 3x and 5x are equivalent.
You can combine like terms – terms with the same variable and power.
Like terms: 4a and 7a are like terms. 4a and 4b are not like terms.
To collect like terms, add or subtract the coefficients: 4a + 7a = 11a.

Solving one-step equations

An equation has an equals sign, like x + 3 = 10.
To solve, get the variable alone on one side.
For x + p = q, subtract p from both sides: x = q − p.
For px = q, divide both sides by p: x = q ÷ p.
Check your answer by putting it back into the equation.

Using formulas

A formula is a rule that shows how two or more things are linked.
For example, the area of a rectangle is A = l × w (A = area, l = length, w = width).
To use a formula, replace the letters with numbers and calculate.
If l = 5 and w = 3, then A = 5 × 3 = 15.
You can also write your own rule to connect two variables, like c = 2n for the cost of n pencils at 2 dollars each.

Special numbers and expressions

Some numbers, like π (π), are not algebraic. They are called transcendental.
An algebraic expression can have powers, like x² or y³.
A square root is a power of 1/2, so √(x) is also an algebraic expression.
When a coefficient is 1, you can leave it out: 1x² is written as x².
When a power is 1, you can leave it out: 3x¹ is written as 3x.

Checking your work

Always check if your answer makes sense.
For equations, substitute your answer back and see if both sides are equal.
For expressions, try a simple number to see if the expression is correct.
If you get a different answer, look for mistakes in signs or order of operations.

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Übungsfragen

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1.What is a letter that stands for an unknown number called?
Easy
- Aa variable
- Ba constant
- Ca coefficient
- Da term
2.In the expression 3n + 5, which part is the variable?
Easy
- An
- B3
- C5
- D+
3.Which of these is an algebraic expression?
Easy
- A3x + 2
- B5 = 3 + 2
- C7 > 2
- D4 + 6
4.In the expression 4y - 7, the constant is 4.
Easy
True or false?
5.Which of the following expressions are equivalent to 2n + 3n? (Select all that apply)
Easy
- A5n
- B6n
- Cn + 4n
- D2n × 3n
- E3n + 2n
6.Solve for x: x + 9 = 17.
Medium
- Ax = 8
- Bx = 26
- Cx = 9
- Dx = 7
7.The expression 3x + 2x simplifies to 5x.
Easy
True or false?
8.Match each term to its meaning.
Medium
- variable
- coefficient
- constant
- a letter for an unknown number
- the number multiplying a variable
- a number without a variable

Mehr Fragen

9.Put these steps in order to solve the equation 3x + 2 = 11.
Medium
- Subtract 2 from both sides
- Divide both sides by 3
- Simplify to get x = 3
10.Which expression is equivalent to 2(x + 3)?
Medium
- A2x + 6
- B2x + 3
- Cx + 6
- D2x + 3x
11.What is the value of the expression 4a - 2 when a = 3?
Easy
- A10
- B14
- C9
- D12
12.The equation x + 5 = 12 has the solution x = 7.
Easy
True or false?
13.Which of the following are like terms with 3x? (Select all that apply)
Medium
- A5x
- B3y
- C-2x
- D3
- Ex²
14.Which of the following is the correct simplification of 5x + 2 - 3x + 4?
Medium
- A2x + 6
- B8x + 6
- C2x + 2
- D8x - 2
15.The expressions 2(x + 1) and 2x + 2 are equivalent.
Medium
True or false?
16.What is the coefficient of x in the expression 7x - 3?
Easy
- A7
- B-3
- Cx
- D3
17.In the expression 2x², the exponent is 2.
Easy
True or false?
18.Which of the following are valid ways to write the expression 'the product of 3 and a number'? (Select all that apply)
Medium
- A3n
- B3 × n
- C3 + n
- D3n²
- En × 3
19.Match each expression to its simplified form.
Medium
- 2x + 3x
- 4x - x
- x + x + x
- 5x
- 3x
- 3x
20.Put these steps in order to evaluate the expression 2x + 3 when x = 5.
Medium
- Substitute x = 5
- Multiply 2 by 5
- Add 3 to get 13
21.What is the value of 2³ + 1?
Medium
- A9
- B7
- C8
- D6
22.The expression 5 - 2x is the same as 5 + (-2x).
Easy
True or false?
23.Which of the following is equivalent to 3(x + 2) - 2x?
Medium
- Ax + 6
- B5x + 6
- Cx + 2
- D3x + 6
24.The equation 3x = 12 has the solution x = 4.
Medium
True or false?
25.Which of the following are equivalent to 2x + 4? (Select all that apply)
Medium
- A2(x + 2)
- Bx + x + 4
- C2x + 2
- D2(x + 4)
- E4 + 2x
26.Match each equation to its solution.
Medium
- x + 5 = 12
- 2x = 14
- x - 3 = 9
- x = 7
- x = 7
- x = 12
27.Put these steps in order to solve the equation 4x - 6 = 10.
Medium
- Add 6 to both sides
- Divide both sides by 4
- Simplify to get x = 4
28.What is the exponent in the expression 5³?
Easy
- A3
- B5
- C15
- D8
29.The expression 3x + 2y has two terms.
Easy
True or false?
30.Match each expression to its number of terms.
Medium
- 3x
- 2x + 1
- x² + 2x + 3
- 1 term
- 2 terms
- 3 terms
31.Put these steps in order to simplify 3x + 2 - x + 4.
Medium
- Combine x terms: 3x - x = 2x
- Combine constants: 2 + 4 = 6
- Write the simplified expression: 2x + 6
32.What is the value of 3²?
Easy
- A9
- B6
- C5
- D8
33.In the expression 4x, the coefficient is 4.
Easy
True or false?
34.Match each word to its meaning.
Easy
- variable
- coefficient
- constant
- term
- a letter that stands for an unknown number
- a number multiplying a variable
- a fixed number that does not change
- a part of an expression separated by + or -
35.Which of the following are algebraic expressions? (Select all that apply)
Easy
- A3x + 2
- B7 = 4 + 3
- C5y
- Da - b
- E2 + 3 = 5
36.In the expression 2x + 3, the number 2 is called the coefficient.
Easy
True or false?
37.Solve for x: 3x = 21.
Medium
- Ax = 7
- Bx = 18
- Cx = 63
- Dx = 24
38.Which of the following are equivalent to 2(x + 4)? (Select all that apply)
Medium
- A2x + 8
- B2x + 4
- Cx + 4 + x + 4
- D2x + 2 × 4
- Ex + 8
39.If n represents a number, what does the expression 2n + 3 mean?
Easy
- ADouble the number and add 3
- BAdd 2 and 3 to the number
- CMultiply the number by 2 and then by 3
- DAdd 2 to the number and then multiply by 3
40.Evaluate the expression 3x - 4 when x = 5.
Medium
- A11
- B15
- C19
- D3
41.Complete the sentence.
Easy
In the expression 5a + 3, the coefficient is ____ and the constant is ____.
42.Evaluate the expression 2x + 7 when x = 4.
Easy
43.Solve for y: 6y = 42.
Medium
44.Write the rule for the area of a rectangle using variables.
Easy
If A stands for area, l for length, and w for width, then A = ____ × ____.
45.Write an expression for 'three times a number plus five'.
Medium
46.A rectangle has length 8 cm and width 5 cm. Use the formula A = l × w to find its area.
Medium
47.Complete the expression: 2n + 3n = ___n.
Easy
2n + 3n = ____n
48.Write an expression for 'a number divided by 4'.
Medium
49.If a = 2 and b = 3, evaluate the expression 4a + 2b - 5.
Medium
50.Complete the equation: x + ___ = 15, if x = 9.
Easy
x + ____ = 15
51.Simplify the expression by combining like terms.
Medium
4y + 2y - 3 = ____
52.Write an equation for: 'The sum of a number and 8 is 20'.
Medium
Let n be the number. The equation is ____.
53.Explain why the expression 2x + 3y cannot be simplified further.
Hard
54.Explain how to check if x = 5 is a solution to the equation 2x + 3 = 13.
Hard
55.Evaluate the expression 2x + 7 when x = 4.
Easy
- A17
- B16
- C15
- D14
56.Solve for y: 6y = 42.
Medium
- A6
- B8
- C7
- D9
57.A rectangle has length 8 cm and width 5 cm. Use the formula A = l × w to find its area.
Medium
- A48 cm²
- B40 cm²
- C36 cm²
- D44 cm²
58.If a = 2 and b = 3, evaluate the expression 4a + 2b - 5.
Medium
- A10
- B11
- C9
- D8

Lernnotizen

## Algebraic Expressions, Equations and Formulae ### What is an algebraic expression? An **algebraic expression** is a maths sentence with numbers, letters, and operations like +, −, ×, ÷. It has no equals sign. For example, `3n + 5` is an expression. The letter `n` stands for an unknown number. ### Parts of an expression - **Variable**: a letter that stands for a number (like `x` or `y`). - **Coefficient**: the number in front of a variable (in `3n`, the coefficient is 3). - **Term**: a single part of an expression, separated by + or − (in `3n + 5`, the terms are `3n` and `5`). - **Constant**: a term with no variable (like `5`). ### Evaluating expressions To **evaluate** an expression, replace the variable with a number and calculate. For example, evaluate `3n + 5` when `n = 2`: `3 × 2 + 5 = 6 + 5 = 11`. ### Equivalent expressions and like terms - **Equivalent expressions** have the same value for any variable. For example, `2x + 3x` is equivalent to `5x`. - **Like terms** have the same variable and power. You can add or subtract them. Example: `4a + 2a = 6a`, but `4a + 2b` cannot be simplified. ### Equations An **equation** is a maths sentence with an equals sign, like `x + 3 = 7`. To solve it, find the value of the variable that makes it true. - For `x + p = q`, subtract `p` from both sides: `x = q - p`. - For `px = q`, divide both sides by `p`: `x = q ÷ p`. Always check your answer by putting it back into the equation. ### Formulae A **formula** is a rule that links two or more variables. For example, the area of a rectangle is `A = l × w` (A = area, l = length, w = width). If `l = 5` and `w = 3`, then `A = 5 × 3 = 15`. ### Writing rules You can write a rule using letters. For example, if you earn 5 dollars per hour, your pay `P` after `h` hours is `P = 5h`. ### Quick tips - A coefficient of 1 is not written: `1x` is just `x`. - A power of 1 is not written: `x^1` is just `x`. - `x^0` is always 1. Keep practising, and you'll be ready for the test!

algebraic expression: A maths sentence with numbers, letters, and operations like +, −, ×, ÷, but no equals sign. Example: 3n + 5.
variable: A letter that stands for an unknown number, like x or y.
coefficient: The number in front of a variable. In 3n, the coefficient is 3.
term: A single part of an expression, separated by + or −. In 3n + 5, the terms are 3n and 5.
constant: A term with no variable, like the 5 in 3n + 5.
evaluate: To replace a variable with a number and calculate the value. For example, evaluate 3n + 5 when n = 2 gives 11.
equivalent expressions: Expressions that have the same value for any variable. For example, 2x + 3x is equivalent to 5x.
like terms: Terms that have the same variable and power, so they can be added or subtracted. Example: 4a and 2a are like terms.
equation: A maths sentence with an equals sign, like x + 3 = 7. Solving it means finding the value of the variable.
formula: A rule that links two or more variables, like A = l × w for the area of a rectangle.

Karteikarten

What is a variable?

A variable is a letter that stands for an unknown number. For example, in 3n + 5, n is a variable.

What is a coefficient?

A coefficient is the number multiplied by a variable. In 3n, the coefficient is 3.

What is a term?

A term is a single number, variable, or product of numbers and variables. In 3n + 5, the terms are 3n and 5.

What is a constant?

A constant is a number that does not change. In 3n + 5, the constant is 5.

Evaluate 3n + 5 when n = 4.

Replace n with 4: 3×4 + 5 = 12 + 5 = 17.

Evaluate 2x - 3 when x = 7.

Replace x with 7: 2×7 - 3 = 14 - 3 = 11.

What does it mean to substitute a value into an expression?

It means replacing the variable with a number and then calculating the result. For example, in 5a, if a = 2, then 5×2 = 10.

Simplify: 3x + 2x

Add the coefficients: 3 + 2 = 5, so 3x + 2x = 5x.

Simplify: 4y + 3 - 2y + 1

Combine like terms: 4y - 2y = 2y, and 3 + 1 = 4, so the answer is 2y + 4.

What are like terms?

Like terms have the same variable raised to the same power. For example, 3x and 5x are like terms, but 3x and 3y are not.

Solve for x: x + 7 = 12

Subtract 7 from both sides: x = 12 - 7, so x = 5.

Solve for y: y - 3 = 9

Add 3 to both sides: y = 9 + 3, so y = 12.

Solve for x: 4x = 20

Divide both sides by 4: x = 20 ÷ 4, so x = 5.

Solve for n: n/3 = 6

Multiply both sides by 3: n = 6 × 3, so n = 18.

How do you check if a number is a solution to an equation?

Plug the number into the equation. If both sides are equal, it is a solution. For example, check x = 5 in 4x = 20: 4×5 = 20, so yes.

What is a formula?

A formula is a rule that shows the relationship between two or more quantities. For example, the area of a rectangle is A = l × w.

Write the formula for the area of a rectangle.

A = l × w, where A is area, l is length, and w is width.

Use the formula A = l × w to find the area of a rectangle with length 8 cm and width 5 cm.

A = 8 × 5 = 40 square cm.

What is an algebraic expression?

An algebraic expression is a combination of numbers, variables, and operations like +, -, ×, ÷. It does not have an equals sign. Example: 3x + 2.

What is an algebraic equation?

An algebraic equation has an equals sign and shows that two expressions are equal. Example: 3x + 2 = 11.

In the expression 5x² + 3x - 2, what is the coefficient of x²?

The coefficient of x² is 5.

In the expression 5x² + 3x - 2, what is the constant term?

The constant term is -2.

Write an expression for 'three times a number plus five'.

Let n be the number. The expression is 3n + 5.

Write an equation for 'a number minus 4 equals 10'.

Let x be the number. The equation is x - 4 = 10.

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