system of equations word problems 3-9 online exercise for | Live ... - Free Printable

Let's solve each of the word problems step by step. These are all systems of linear equations problems, so we'll define variables, write equations based on the information, and solve them.

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Problem 1:

> Find two numbers if their sum is 19 and their difference is 5.

#### Step 1: Define variables
Let:
- $ x $ = first number
- $ y $ = second number

#### Step 2: Write equations
From the problem:
1. $ x + y = 19 $ (sum is 19)
2. $ x - y = 5 $ (difference is 5)

#### Step 3: Solve the system
Add the two equations:
$$
(x + y) + (x - y) = 19 + 5 \\
2x = 24 \\
x = 12
$$

Substitute $ x = 12 $ into equation 1:
$$
12 + y = 19 \\
y = 7
$$

✔ Answer: The numbers are 12 and 7.

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Problem 2:

> The difference of two numbers is 9. Their sum is 33. Find the numbers.

#### Step 1: Define variables
Let:
- $ x $ = larger number
- $ y $ = smaller number

#### Step 2: Write equations
1. $ x - y = 9 $
2. $ x + y = 33 $

#### Step 3: Solve the system
Add the equations:
$$
(x - y) + (x + y) = 9 + 33 \\
2x = 42 \\
x = 21
$$

Substitute into equation 2:
$$
21 + y = 33 \\
y = 12
$$

✔ Answer: The numbers are 21 and 12.

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Problem 3:

> For the play, Georgie sold 3 adult tickets and 1 child ticket for $44. The next day she sold 3 adult tickets and 4 child tickets for $68. Find the price of an adult ticket and a child ticket.

#### Step 1: Define variables
Let:
- $ a $ = price of an adult ticket
- $ c $ = price of a child ticket

#### Step 2: Write equations
1. $ 3a + 1c = 44 $ (first day)
2. $ 3a + 4c = 68 $ (second day)

#### Step 3: Solve the system
Subtract equation 1 from equation 2:
$$
(3a + 4c) - (3a + c) = 68 - 44 \\
3a + 4c - 3a - c = 24 \\
3c = 24 \\
c = 8
$$

Now substitute $ c = 8 $ into equation 1:
$$
3a + 8 = 44 \\
3a = 36 \\
a = 12
$$

✔ Answer: Adult ticket = $12, Child ticket = $8

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Problem 4:

> For the senior class trip, the 240 students rented 8 vans and 8 buses. At another school, the 54 seniors rented 4 vans and 1 bus. Every van carried the same number of students and every bus carried the same number of students. How many students were in each van and in each bus?

#### Step 1: Define variables
Let:
- $ v $ = number of students per van
- $ b $ = number of students per bus

#### Step 2: Write equations
1. $ 8v + 8b = 240 $ (first group)
2. $ 4v + 1b = 54 $ (second group)

#### Step 3: Simplify and solve
Simplify equation 1:
$$
8v + 8b = 240 \Rightarrow v + b = 30 \quad \text{(divide both sides by 8)}
$$

So now we have:
1. $ v + b = 30 $
2. $ 4v + b = 54 $

Subtract equation 1 from equation 2:
$$
(4v + b) - (v + b) = 54 - 30 \\
4v + b - v - b = 24 \\
3v = 24 \\
v = 8
$$

Now plug into equation 1:
$$
8 + b = 30 \\
b = 22
$$

✔ Answer: Each van carried 8 students, each bus carried 22 students.

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✔ Final Answers Summary:

1. Numbers: 12 and 7
2. Numbers: 21 and 12
3. Adult ticket: $12, Child ticket: $8
4. Van: 8 students, Bus: 22 students

Let me know if you'd like these explained with diagrams or tables!