Deriving and Evaluating Expressions with One Variable
Expressions with one variable combine numbers with a letter, like 20 + 5x. Here 20 is the starting amount and 5x adds 5 for each x. To evaluate it, replace the letter with a number: when x = 8, 20 + 5 × 8 = 60.
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Understanding Variables
- A variable is a letter that stands for a number that can change.
- For example, x can represent the number of months passed.
Understanding Expressions
- An expression is formed by combining numbers and variables.
- It shows how a value changes and has no equals sign.
Forming Expressions
- Example 1: 20 + 5x means start with 20 and add 5 for each x.
- Example 2: 50 − 3y means start with 50 and subtract 3 for each y.
Evaluating Expressions
- To evaluate an expression, you replace the variable with a number.
- You then calculate to find the value of the expression.
Practice Questions
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Julia has £20 in savings. Each month, she saves an additional £5. How much will she have in savings in total after 5 months?
Correct! 🎉 +10 pointsCorrect! 🎉 Worth 10 XPNot quite right
The expression is , where x is the number of months.
Multiplication comes before addition, so after 5 months Julia has (£) in savings in total.
A library has 200 books. Every week, 10 new books are added. How many books will the library have after 3 weeks?
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The expression is , where x is the number of weeks.
Multiplication comes before addition, so after 3 weeks the library has books.
You start with £100, and you spend £8 each day. How much money will you have left after 12 days?
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The expression is , where x is the number of days.
In 12 days you spend , so you have (£) left.
A factory starts with 1000 units of a product. They produce 125 units every day. After how many days will they have a total of 2000 units?
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The equation is , where x is the number of days.
Subtract 1000 from both sides: . Then divide both sides by 125: days.
A gardener plants 15 flowers each day. They start with 50 flowers already planted. How many flowers will they have after 7 days?
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The expression is , where x is the number of days.
Multiplication comes before addition, so after 7 days the gardener has flowers in total.
A photographer charges £50 for a session. They charge an additional £30 for each photo after the first. How much will the photographer charge after 12 photos?
Correct! 🎉 +30 pointsCorrect! 🎉 Worth 30 XPNot quite right
The first photo is covered by the £50 session fee, so only the photos after the first cost £30 each. The expression is .
With 12 photos, , so there are photos to charge for: (£).
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Interactive Activity
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Frequently asked questions
The questions students bump into most on this topic
An expression is a combination of numbers and letters joined by operations, with no equals sign. 20 + 5x is an expression: it means start with 20 and add 5 for every x. Other examples are 50 − 3y, 5x and 3 + 2. An expression does not have an answer on its own; it takes a value once you know what the letter stands for.
A variable is a letter, usually x or y, that stands for a number that can change; it is called a variable because its value can vary. In Julia’s savings, 20 + 5x, the x is the number of months: it can be 8, 12 or 20, and the expression gives her savings for each. One rule replaces a new sum every time.
An expression has no equals sign; an equation does. 20 + 5x is an expression: you can evaluate it for any value of x, but you cannot solve it. 20 + 5x = 60 is an equation: it says the expression equals 60, and you can solve it to find x = 8. An equation is two expressions with an equals sign between them.
5x means 5 times x. When a number is written directly in front of a letter, you multiply them; the times sign is left out. If x is 8, then 5x is 5 × 8 = 40, so Julia’s savings 20 + 5x come to 20 + 40 = £60. In the same way 3y means 3 times y, and x on its own means 1x.
Replace the letter with the number, then work it out, doing the multiplication before the addition. For 20 + 5x with x = 8: 20 + 5 × 8 = 20 + 40 = 60. For 50 − 3y with y = 10: 50 − 3 × 10 = 50 − 30 = 20. Exam papers call this evaluating the expression.
Find the fixed amount and the amount that repeats, then use a letter for the number of repeats. Julia starts with £20 and saves £5 a month: after x months she has 20 + 5x. You have £50 and each pack of snacks costs £3: after buying y packs you have 50 − 3y left. Starting amount, then plus or minus the repeated amount times the letter.
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