https://r2.erweima.ai/mathbot/image/1725968293860g2wvyg5w.jpg

I'll help you solve the problems from the image. Let's break down and solve each of the tasks:

### Aufgabe 1: Bestimme die Lösung (Linear Equations)

#### a) \( 6x + 14 = 38 \)
\[
6x = 38 - 14 \implies 6x = 24 \implies x = \frac{24}{6} = 4
\]
**Solution:** \( x = 4 \)

#### b) \( 7x - 9 = 4x + 6 \)
\[
7x - 4x = 6 + 9 \implies 3x = 15 \implies x = \frac{15}{3} = 5
\]
**Solution:** \( x = 5 \)

#### c) \( 5 + 7x = 2x - 11 + 3x \)
\[
7x - 5x = -11 - 5 \implies 2x = -16 \implies x = \frac{-16}{2} = -8
\]
**Solution:** \( x = -8 \)

#### d) \( 12 + 8x = 5x - 12 \)
\[
8x - 5x = -12 - 12 \implies 3x = -24 \implies x = \frac{-24}{3} = -8
\]
**Solution:** \( x = -8 \)

#### e) \( 3(x + 2) - 5(x - 1) = 11 \)
\[
3x + 6 - 5x + 5 = 11 \implies -2x + 11 = 11 \implies -2x = 0 \implies x = 0
\]
**Solution:** \( x = 0 \)

#### f) \( 3(x + 4) + 4x + 3(x - 11) = 6(2x - 3) - 18 \)
\[
3x + 12 + 4x + 3x - 33 = 12x - 18 - 18 \\
10x - 21 = 12x - 36 \\
-2x = -15 \implies x = \frac{15}{2} = 7.5
\]
**Solution:** \( x = 7.5 \)

#### g) \( \frac{3}{4}(8x + 4) = 2 \left( 3x - \frac{5}{2} \right) + x \)
\[
\frac{3}{4}(8x + 4) = 6x - 5 + x \implies 6x + 3 = 7x - 5 \\
6x - 7x = -5 - 3 \implies -x = -8 \implies x = 8
\]
**Solution:** \( x = 8 \)

---

### Aufgabe 2: Rechteckiges Grundstück

- Umfang: \( 72 \, \text{m} \)
- Eine Seite ist 6 m länger als die andere.

Let the shorter side be \( x \) and the longer side be \( x + 6 \).

The perimeter formula is \( 2x + 2(x + 6) = 72 \).
\[
2x + 2x + 12 = 72 \implies 4x + 12 = 72 \implies 4x = 60 \implies x = 15
\]
The sides are 15 m and \( 15 + 6 = 21 \, \text{m} \).

#### a) **Seitenlängen:** 15 m and 21 m.

#### b) **Flächeninhalt:** \( 15 \times 21 = 315 \, \text{m}^2 \).

---

### Aufgabe 3: Rechteck mit Umfang 60 cm

- Eine Seite ist dreimal so lang wie die andere.

Let the shorter side be \( x \) and the longer side be \( 3x \).

Perimeter equation: \( 2x + 2(3x) = 60 \)
\[
2x + 6x = 60 \implies 8x = 60 \implies x = \frac{60}{8} = 7.5
\]
The shorter side is 7.5 cm, and the longer side is \( 3 \times 7.5 = 22.5 \, \text{cm} \).

#### **Seitenlängen:** 7.5 cm and 22.5 cm.

---

### Aufgabe 4: Andre's Pocket Money

- Total pocket money: \( 60 \, \text{€} \)
- 2nd day: Spent 3 times what was spent on the 1st day.
- 3rd day: Spent 3 € less than the 1st day.
- Brought back 28 €.

Let the amount spent on the first day be \( x \).

Total spent over 3 days:
\[
x + 3x + (x - 3) = 60 - 28 = 32
\]
\[
5x - 3 = 32 \implies 5x = 35 \implies x = 7
\]
- **1st day:** 7 €
- **2nd day:** \( 3 \times 7 = 21 \, \text{€} \)
- **3rd day:** \( 7 - 3 = 4 \, \text{€} \)

---

### Aufgabe 5: Fruit Prices

- 8 bananas, 10 apples, 3 oranges total: \( 6.20 \, \text{€} \)
- 1 apple costs 10 cents less than a banana.
- 1 orange costs 30 cents more than a banana.

Let the cost of a banana be \( x \).

The total cost equation:
\[
8x + 10(x - 0.10) + 3(x + 0.30) = 6.20
\]
\[
8x + 10x - 1 + 3x + 0.90 = 6.20 \implies 21x - 0.10 = 6.20 \implies 21x = 6.30 \implies x = \frac{6.30}{21} = 0.30
\]
- **Banana:** 0.30 €
- **Apple:** \( 0.30 - 0.10 = 0.20 \, \text{€} \)
- **Orange:** \( 0.30 + 0.30 = 0.60 \, \text{€} \)

---

Let me know if you'd like more details, or if you have any questions.

**5 Related Questions:**
1. How do you solve a system of linear equations?
2. What is the perimeter formula for a rectangle?
3. How does the equation of a line change based on slope and intercept?
4. What are the typical methods for calculating the area of geometric figures?
5. How can you calculate the price of individual items based on total price constraints?

**Tip:** Always check your solution by substituting the values back into the original equation to verify they satisfy the condition.

Solve linear equations and word problems about the perimeter, area, and prices. Learn how to calculate the sides of rectangles and solve real-world math problems involving pocket money and shopping.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation