Introduction to Formulas
A formula is a maths rule that links two variables, like y = 2x + 3. Whichever value you pick for x, the rule gives one matching y. To find it, substitute your value in place of x and work it out: if x = 1, then y = 2 × 1 + 3, so y = 5.
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What Is a Formula?
- A formula is a mathematical rule that links two quantities.
- You substitute x values into a formula to find the corresponding y values.
How to Visualise a Formula?
- Use the formula to make a table of x and y values.
- Then plot these values on a graph to see the pattern.
Using Formulas to Solve Problems
- A formula can represent real-life situations like saving money.
- You substitute values to solve specific problems.
Practice Questions
Test your understanding
If , what is y when ?
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Substitute into the formula. Remember means , and the multiplication is done before the addition.
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The formula for the distance travelled by a car is , where y is the distance in miles and x is the number of hours driven. How far will it travel after 3 hours?
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Substitute into the formula: .
The car travels 300 miles. Note that and are unlike terms, so they cannot be combined into before substituting, and the must still be added after working out .
If , what is x when ?
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Substitute : .
Add 7 to both sides (the inverse of subtracting 7): .
Divide both sides by 4 (the inverse of multiplying by 4): .
A company’s total revenue y is represented by , where x is the number of products sold. How many products must be sold to reach a revenue of £4800?
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Substitute : .
Add 200 to both sides (the inverse of subtracting 200): .
Divide both sides by 500: .
So 10 products must be sold.
The total cost of a gym membership is calculated by , where x is the number of months and y is the total cost in pounds. If the total cost is £200, how many months did the member sign up for?
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Substitute , because £200 is the total cost: .
Subtract 50 from both sides: .
Divide both sides by 30: .
The member signed up for 5 months.
A company charges a £200 setup fee and £50 for printing each book. If you print more than 10 books, you get a discount on the total cost. How many books can be printed for £720?
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For more than 10 books there is a discount, so you pay of the full cost: .
Set : .
Divide both sides by 0.9: .
Subtract the £200 setup fee from both sides: .
Divide both sides by 50: .
Since , the discount really does apply, so 12 books can be printed.
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Frequently asked questions
The questions students bump into most on this topic
A formula is a rule, written as an equation, that links two quantities: if you know one, it tells you the other. y = 2x + 3 is a formula. When x is 1, y is 2 × 1 + 3 = 5; when x is 2, y is 7; when x is 3, y is 9. “Formulas” and “formulae” are both correct plurals.
A mapping shows how two quantities are related, often with an arrow: countries are mapped to their capitals. A formula is a mapping written as a mathematical equation, so y = 2x + 3 maps each x to its matching y. Every formula is a mapping; a formula is the kind you can calculate with.
Replace the letter with the value you know, then work it out. For y = 2x + 3 with x = 3: y = 2 × 3 + 3 = 9. For a car whose distance is y = 60x + 120 after x hours, 3 hours gives y = 60 × 3 + 120 = 300 miles. Multiply before you add, and keep the units in mind.
First make a table of values: choose some x values and use the formula to find each y. For y = 2x + 3, x = 1, 2, 3 gives y = 5, 7, 9. Then plot each pair as a point, with x across and y up. For this formula all the points fall on a straight line, which shows the pattern at a glance.
Turn the situation into a formula, then substitute or use a table. Shoes cost £80; you have £20 saved and save £12 a week. Your savings after x weeks are y = 20 + 12x. You need y = 80: try values in a table, and when x = 5, y = 20 + 12 × 5 = 80. So the shoes take 5 weeks.
A formula links two or more variables with a rule you use again and again: y = 2x + 3 gives a y for every x. An equation is true only for particular values and is there to be solved: 2x + 3 = 9 has the one answer x = 3. An expression such as 2x + 3 has no equals sign. GCSE papers ask you to tell the three apart.
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