https://file.aiquickdraw.com/mathbot/file/17254389257990r14lbdw.jpg

To solve these equations, I will first extract and interpret the equations from the image. I'll then proceed to solve them one by one for the variable \( x \). Here are the equations as extracted from the image:

1. \(\frac{x + 3}{7} = 1\)
2. \(x - 3.1 = 4\)
3. \(\frac{5x}{8} = 4\)
4. \(4 + 2x = 8\)
5. \(2x + 4 = 12\)
6. \(3x - 7 = 2x + 1\)
7. \(\frac{2x - 4}{5} = 7\)
8. \(2x - 8 = 3(x + 1)\)
9. \(\frac{x - 1}{2} - \frac{x + 1}{3} = \frac{1}{6}\)
10. \(\frac{2x + 3}{x - 1} = \frac{3}{2}\)

I'll now solve each equation for \( x \):

1. **\(\frac{x + 3}{7} = 1\)**
   \[
   x + 3 = 7 \quad \text{(Multiplying both sides by 7)}
   \]
   \[
   x = 7 - 3 = 4
   \]

2. **\(x - 3.1 = 4\)**
   \[
   x = 4 + 3.1 = 7.1
   \]

3. **\(\frac{5x}{8} = 4\)**
   \[
   5x = 4 \times 8 = 32 \quad \text{(Multiplying both sides by 8)}
   \]
   \[
   x = \frac{32}{5} = 6.4
   \]

4. **\(4 + 2x = 8\)**
   \[
   2x = 8 - 4 = 4
   \]
   \[
   x = \frac{4}{2} = 2
   \]

5. **\(2x + 4 = 12\)**
   \[
   2x = 12 - 4 = 8
   \]
   \[
   x = \frac{8}{2} = 4
   \]

6. **\(3x - 7 = 2x + 1\)**
   \[
   3x - 2x = 1 + 7
   \]
   \[
   x = 8
   \]

7. **\(\frac{2x - 4}{5} = 7\)**
   \[
   2x - 4 = 7 \times 5 = 35
   \]
   \[
   2x = 35 + 4 = 39
   \]
   \[
   x = \frac{39}{2} = 19.5
   \]

8. **\(2x - 8 = 3(x + 1)\)**
   \[
   2x - 8 = 3x + 3
   \]
   \[
   2x - 3x = 3 + 8
   \]
   \[
   -x = 11 \quad \Rightarrow \quad x = -11
   \]

9. **\(\frac{x - 1}{2} - \frac{x + 1}{3} = \frac{1}{6}\)**
   \[
   3(x - 1) - 2(x + 1) = 1 \quad \text{(Multiply the equation by 6)}
   \]
   \[
   3x - 3 - 2x - 2 = 1
   \]
   \[
   x - 5 = 1 \quad \Rightarrow \quad x = 6
   \]

10. **\(\frac{2x + 3}{x - 1} = \frac{3}{2}\)**
    Cross-multiply:
    \[
    2(2x + 3) = 3(x - 1)
    \]
    Expand and simplify:
    \[
    4x + 6 = 3x - 3
    \]
    \[
    x = -9
    \]

Would you like further details on any of these solutions? Here are some related questions to consider:

1. How would you solve a similar equation involving fractions?
2. What happens if the variable \( x \) appears in the denominator?
3. How do you solve an equation that has variables on both sides?
4. How do you check if your solution to an equation is correct?
5. What strategies can be used when solving complex algebraic equations?

**Tip:** Always double-check your solution by substituting it back into the original equation to verify its correctness.

Explore the process of solving different types of algebraic equations, including linear and rational equations. This problem set provides step-by-step solutions for finding x, covering key algebraic concepts such as linear equations, rational equations, and the use of cross-multiplication. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation