Algebraic expressions, equations and formulae
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課程筆記
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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練習題
免費預覽——58 題中的 8 題。註冊即可查看全部。
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.
EasyTrue 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.
EasyTrue 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
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