Systems of Equations Word Problems

Show your work for each problem.

1.
The sum of two numbers is 20 and their difference is 6. Which system of equations represents this situation?
1. a) and
2. b) and
3. c) and
4. d) and
Answer: and
Sum of two numbers is . Difference means (larger minus smaller).
2.
The sum of two numbers is 30 and their difference is 10. What is the larger number?
Answer: 20
Adding the equations: , so . The larger number is 20 (and the smaller is 10).
3.
Adult tickets cost 10 euros and child tickets cost 6 euros. A family bought 5 tickets for 38 euros. How many adult tickets did they buy?
1. a)4
2. b)3
3. c)1
4. d)2
Answer: 2
From : . Substitute: , so , , .
4.
The sum of two numbers is 45. One number is 9 more than the other. Find both numbers.
Answer: 27, 18
1. Write the sum equation: 45
2. If is the larger number and it's 9 more than , write: 9
3. Substitute into the sum: . Solve for . 18
4. Now find 27
5.
A cafe sold 80 drinks in one hour. Coffee costs 4 euros and tea costs 3 euros. Total sales were 290 euros. How many coffees were sold?
Answer: 50
From : . Substitute: , so , .
6.
A boat travels 36 km downstream in 2 hours and 36 km upstream in 3 hours. What is the speed of the boat in still water?
1. a)15 km/h
2. b)6 km/h
3. c)18 km/h
4. d)12 km/h
Answer: 15 km/h
Downstream speed: km/h = . Upstream speed: km/h = . Adding: , so km/h.
7.
Maria has 30 coins. Some are 2-euro coins and some are 50-cent coins. The total value is 42 euros. How many of each coin does she have?
Answer: 18, 12
1. Let = 2-euro coins, = 50-cent coins. Write the coin count equation: 30
2. Write the value equation in euros: 42
3. Multiply the value equation by 2 to eliminate decimals: 84
4. Subtract: . Solve for . 18
5. Find 12
8.
Twice a number plus three times another number equals 31. The first number minus the second equals 3. What is the first number?
Answer: 8
Let = first number, = second number. From : . Substitute into : , so , , . Therefore . Verify: ✓, ✓.
9.
A chemist needs to mix a 20% acid solution with a 50% acid solution to get 600 mL of a 30% solution. How much of the 20% solution is needed?
1. a)200 mL
2. b)300 mL
3. c)500 mL
4. d)400 mL
Answer: 400 mL
Let = 20% solution, = 50% solution. and . From first: . Sub: , so , , mL.
10.
A store sold 45 shirts. Small shirts cost 15 euros and large shirts cost 22 euros. Total sales were 780 euros. How many of each size were sold?
Answer: 30, 15
1. Let = small, = large. Write the quantity equation: 45
2. Write the revenue equation: 780
3. From equation 1: . Substitute into equation 2: . Simplify: . Solve for . 15
4. Find 30
11.
A plane flies 2400 km with the wind in 3 hours. The return trip against the wind takes 4 hours. What is the speed of the wind in km/h?
Answer: 100
With wind: . Against wind: . Subtracting: , so km/h.
12.
Three times a number plus twice another equals 30. The first number minus twice the second equals 2. What is the sum of the two numbers?
1. a)15
2. b)11
3. c)9
4. d)13
Answer: 11
Add the equations: , so , . From : , , . Sum = . Verify: ✓.
13.
A farmer has chickens and cows. Together they have 50 heads and 140 legs. How many chickens and cows does the farmer have?
Answer: 30, 20
1. Let = chickens (2 legs each), = cows (4 legs each). Write the heads equation: 50
2. Write the legs equation: 140
3. Simplify the legs equation by dividing by 2: 70
4. Subtract heads equation from simplified legs: . Solve for . 20
5. Find 30
14.
In 5 years, a father will be twice as old as his son. 3 years ago, the father was 3 times as old as the son. How old is the son now?
Answer: 11
Let = father's age, = son's age. Future: , so . Past: , so . Setting equal: , so . The son is 11 years old. Verify: Father is . In 5 years: ✓. 3 years ago: ✓.
15.
A rectangle has a perimeter of 56 cm. The length is 4 cm more than twice the width. Find the dimensions.
Answer: 20, 8
1. Let = length, = width. Write the perimeter equation: 56
2. Simplify: 28
3. The length is 4 more than twice the width. Write: 4
4. Substitute into : . Solve for . 8
5. Find 20