Year 7 Maths: Solve Word Problems by Finding Two-Variable Equations

Introduction to Two-Variable Equations

Hello, Year 7! Today, we will learn how to solve word problems by finding two-variable equations. A two-variable equation is an equation that contains two different variables (like x and y). These equations help us understand relationships between different things.

What is a Variable?

A variable is a letter that stands for a number we do not know yet. For example, in the equation

Why Use Two-Variable Equations?

We use two-variable equations to describe situations where two things are related. For example, if you have some apples and oranges, you can create an equation to represent the total number you have.

Steps to Solve Word Problems

1. Read the Problem Carefully: Understand what is being asked.
2. Identify the Variables: Decide what the letters (variables) will represent.
3. Create an Equation: Use the information given in the problem to write an equation.
4. Solve the Equation: Find the values of the variables.

Example 1: Simple Problem

Problem: Lucy has 3 apples, and she buys some more apples. If she now has 8 apples, how many did she buy?

• Step 1: Identify the variable. Let $$x$$ be the number of apples Lucy bought.
• Step 2: Write the equation. The equation is $$3 + x = 8$$ .
• Step 3: Solve the equation. Subtract 3 from both sides: $$x = 8 – 3$$ , so $$x = 5$$ .

Answer: Lucy bought 5 apples.

Example 2: More Complex Problem

Problem: A school is planning a trip for students. The cost of the trip is £50 per student. If there are

• Step 1: Identify the variables. Let $$x$$ be the number of students and $$y$$ be the total money needed.
• Step 2: Write the equation. The equation is $$y = 50x$$ .
• Step 3: If there are 20 students, substitute x: $$y = 50(20)$$ , so $$y = 1000$$ .

Answer: They need to raise £1000 for 20 students.

Key Rules, Tips, and Tricks

1. Always define your variables at the beginning.
2. Write down your equations clearly and step by step.
3. Double-check your calculations to avoid mistakes.
4. Practice with different problems to build your confidence.

Questions

Easy Level Questions

1. Sam has 2 candies and buys $$x$$ more. If he has 6 candies now, what is $$x$$ ?
2. Mia has $$y$$ balloons. If she has 5 balloons now, how many does she need to have 10?
3. There are $$x$$ birds on a tree. If 3 more come, there are 8. What is $$x$$ ?
4. Tom has 4 pencils and buys $$x$$ more. If he has 10 pencils now, how many did he buy?
5. A box has $$x$$ chocolates. If it has 7 and you eat 2, how many are left?
6. Lisa has $$y$$ stickers. She has 3 now. How many does she need to have 10?
7. A shop sells $$x$$ toys. If there are 5 toys and 3 more come, how many are there?
8. Anna has 6 marbles and finds $$x$$ more. If she has 10 now, how many did she find?
9. A basket has $$y$$ apples. If it has 4 apples now, how many to have 12?
10. Alex has 3 books and needs $$x$$ more. If he wants 10, how many does he need?
11. There are $$x$$ ducks in a pond. If 2 more come, there are 5. What is $$x$$ ?
12. Jamie has $$y$$ coins. If he has 1 now, how many does he need to have 8?
13. A movie ticket costs £7. If there are $$x$$ tickets, how much is it?
14. Sarah has 5 flowers and buys $$x$$ more. If she has 12 now, how many did she buy?
15. A jar has $$x$$ cookies. If there are 10 and you eat 3, how many are left?
16. A cake has $$y$$ pieces. If it has 6 pieces now, how many to have 15?
17. There are $$x$$ cars in a parking lot. If 4 come in, there are 9. What is $$x$$ ?
18. Ben has 2 video games and buys $$x$$ more. If he has 6 now, how many did he buy?
19. A plant has $$y$$ leaves. If it has 5 leaves now, how many to have 10?
20. A library has $$x$$ books. If there are 7 and 3 more come, how many are there?

Medium Level Questions

1. If $$x$$ students go on a trip and each pays £10, how much money is collected?
2. A teacher has $$y$$ markers. If he buys 5 more, how many does he have now?
3. A car travels $$x$$ miles. If it traveled 50 miles and then 20 more, how far did it go?
4. A store has $$y$$ shirts. If it sells 3 and now has 5, how many were there originally?
5. A recipe needs $$x$$ cups of flour. If you have 2 and need 5, how much more do you need?
6. A train leaves at $$x$$ o’clock. If it leaves 2 hours later, what time does it leave?
7. A book costs £5. If $$x$$ books are bought, how much is spent?
8. A box holds $$y$$ toys. If it has 4 toys and 2 more are added, how many are in the box now?
9. A pizza has $$x$$ slices. If you eat 2 and there are 6 left, how many were there?
10. A gardener plants $$y$$ flowers. If 3 are planted and he wants a total of 15, how many more does he need?
11. A box has $$x$$ crayons. If there are 10 and you take 3, how many are left?
12. A budget is $$y$$ pounds. If you spend 20 and have 30 left, what was your budget?
13. A recipe calls for $$x$$ eggs. If you have 2 and need 5, how many more do you need?
14. A car travels $$y$$ miles. If it has 15 miles left to travel and started with 100, how far has it gone?
15. A shop sells $$x$$ toys. If it has 3 and sells 5, how many are left?
16. A team has $$y$$ players. If 2 are injured, how many are left?
17. A box contains $$x$$ balls. If you add 5 and have 12 now, how many were there before?
18. A plant grows $$y$$ cm. If it grows 2 cm in a week, how tall is it after a month?
19. A movie ticket costs £8. If $$x$$ tickets are sold, how much total money is made?
20. A basket has $$x$$ oranges. If it has 10 now and you buy 5 more, how many will there be?

Hard Level Questions

1. A school has $$x$$ students. If each student pays £50 for a trip, how much money is collected for 200 students?
2. A recipe needs $$x$$ cups of sugar. If you have 4 and need 10, how much more do you need?
3. A train travels $$x$$ miles in an hour. If it travels for 3 hours, how far does it go?
4. A store sells $$y$$ items. If it sells 30 items and has 70 left, how many were there originally?
5. A car has $$x$$ litres of fuel. If it consumes 5 litres per hour, how much fuel does it have after 3 hours?
6. A box contains $$y$$ toys. If 5 are removed and it now contains 10, how many were there originally?
7. A gardener plants $$x$$ flowers. If he has 30 flowers and plants 10 more, how many does he have now?
8. A budget for a party is $$y$$ pounds. If £50 is spent and £150 is left, what was the original budget?
9. A shop has $$x$$ computers. If 15 are sold and there are 25 left, how many were there originally?
10. A car travels $$y$$ miles. If it has 200 miles left and started with 300, how far has it gone?
11. A recipe requires $$x$$ grams of cheese. If you have 200g and need 500g, how much more do you need?
12. A box holds $$y$$ chocolates. If there are 15 chocolates left and 5 are eaten, how many were there before?
13. A plant grows $$x$$ cm each day. If it grows for 7 days, how much will it grow?
14. A store sells $$y$$ games. If it has 25 games and sells 15, how many are left?
15. A cinema has $$x$$ seats. If it sells tickets for 80 seats and 20 are left, how many are there?
16. A team has $$y$$ players. If 5 are injured and there were 15 players originally, how many are left?
17. A box has $$x$$ books. If you buy 3 and now have 20, how many were there before?
18. A recipe needs $$y$$ litres of milk. If you have 1 litre and need 3, how much more do you need?
19. A car travels $$x$$ miles in a week. If it travels for 4 weeks, how far does it go?
20. A budget is $$y$$ pounds. If you spend £100 and have £400 left, what was the original budget?

Answers

Easy Level Answers

1. $$x = 6 – 2 = 4$$ . So, Sam bought 4 candies.
2. $$y = 10 – 5 = 5$$ . Mia needs 5 balloons to have 10.
3. $$x = 8 – 2 = 6$$ . There were 6 birds on the tree.
4. $$x = 10 – 4 = 6$$ . Tom bought 6 pencils.
5. $$x = 10 – 7 = 3$$ . There are 3 cookies left.
6. $$y = 10 – 3 = 7$$ . Lisa needs 7 stickers to have 10.
7. $$x = 5 – 3 = 2$$ . There are now 2 toys in the shop.
8. $$x = 10 – 6 = 4$$ . Anna found 4 marbles.
9. $$y = 12 – 4 = 8$$ . The basket needs 8 more apples.
10. $$x = 10 – 3 = 7$$ . Alex needs 7 more books.
11. $$x = 5 – 2 = 3$$ . There were 3 ducks in the pond.
12. $$y = 8 – 1 = 7$$ . Jamie needs 7 coins to have 8.
13. $$y = 7x$$ . If $$x = 10$$ , total cost = £70.
14. $$x = 5$$ . Sarah bought 5 flowers.
15. $$x = 10 – 3 = 7$$ . There are 7 cookies left.
16. $$y = 15 – 6 = 9$$ . The gardener needs 9 more flowers.
17. $$x = 12 – 10 = 2$$ . There were 2 crayons before.
18. $$y = 400$$ . The budget was £400.
19. $$y = 3 – 1 = 2$$ . The gardener needs 2 more litres.
20. $$x = 10 + 2 = 12$$ . The box had 12 books.

Medium Level Answers

1. Total collected = £10x. If 200 students go, it’s £2000.
2. Total markers = $$y + 5$$ . If he had 10, he has 15 now.
3. Total miles = $$x + 20$$ . If it was 50, it travelled 70 miles.
4. Original shirts = $$y + 30$$ . If he has 5 now, it was 35.
5. More flour needed = $$5 – 2 = 3$$ . You need 3 more cups.
6. Departure time = $$x + 2$$ . If it leaves at 2, it leaves at 4.
7. Total spent = £5x. For 10 books, it’s £50.
8. Total toys = $$y + 5$$ . If it has 10 now, it had 15.
9. Original slices = $$x + 2 + 6$$ . It had 8 slices before.
10. More flowers needed = $$15 – 3 = 12$$ . He needs 12 more flowers.
11. Original crayons = $$x + 3 = 10$$ . There were 10 crayons before.
12. Original budget = $$y + 20 + 30$$ . So, it was £100.
13. More eggs needed = $$5 – 2 = 3$$ . You need 3 more eggs.
14. Total miles travelled = $$100 – 15 = 85$$ . It travelled 85 miles.
15. Original toys = $$x + 3 – 5$$ . So, it had 8 toys.
16. Remaining players = $$y – 5$$ . If there were 15 players, there are 10 left.
17. Original balls = $$x – 5 + 12$$ . It had 17 balls before.
18. Total height = $$x \times 30$$ . It grows 30 cm in a month.
19. Total money = £8x. If 20 tickets are sold, it’s £160.
20. Original oranges = $$x – 5 = 10$$ . It had 15 oranges before.

Hard Level Answers

1. Total collected = $$200 \times 50 = £10000$$ . If each student pays 50, it’s £10000.
2. More sugar needed = $$10 – 4 = 6$$ . You need 6 more cups of sugar.
3. Total distance = $$x \times 3$$ . If it travels for 3 hours, it goes $$3x$$ miles.
4. Original items = $$y + 30$$ . If it has 70 left, it had 100 items originally.
5. Total fuel left = $$x – 5 \times 3$$ . After 3 hours, it’s $$x – 15$$ litres.
6. Original toys = $$y + 5$$ . If it has 10 now, it had 15.
7. Total flowers = $$30 + 10$$ . Now he has 40 flowers.
8. Original budget = $$150 + 50 + 100$$ . So, it was £300.
9. Original computers = $$x + 15 + 25$$ . So, it was 65 computers.
10. Total travelled = $$300 – 200 = 100$$ . It has gone 100 miles.
11. More cheese needed = $$500 – 200 = 300$$ . You need 300g more.
12. Original chocolates = $$15 + 5 = 20$$ . It had 20 chocolates before.
13. Total growth = $$x \times 7$$ . It grows $$7x$$ cm in a week.
14. Remaining games = $$y – 15$$ . If it sold 15, how many are left?
15. Seats remaining = $$x – 80$$ . If 20 are left, there are 100 seats.
16. Remaining players = $$15 – 5 = 10$$ . There are 10 players left.
17. Original books = $$20 – 3 = 17$$ . It had 17 books before.
18. More milk needed = $$3 – 1 = 2$$ . You need 2 more litres.
19. Total distance = $$x \times 4$$ . If it goes $$x$$ miles a week, it travels $$4x$$ miles.
20. Original budget = £500. If you have £400 left from £500, it was £500.

Feel free to practice these problems and remember to take your time. You can do it! Happy learning!