Let's start by exploring five related exercises with solutions. These exercises will cover basic algebra, solving equations, and some more advanced concepts like systems of equations and quadratic equations.

### 1. **Linear Equation - Solve for \(x\):**
   \[
   3x - 5 = 10
   \]

**Solution:**
\[
3x - 5 = 10 \\
3x = 10 + 5 \\
3x = 15 \\
x = \frac{15}{3} = 5
\]

### 2. **Solving for \(y\) in Terms of \(x\):**
   \[
   2x + 3y = 12
   \]

**Solution:**
\[
2x + 3y = 12 \\
3y = 12 - 2x \\
y = \frac{12 - 2x}{3} = 4 - \frac{2x}{3}
\]

### 3. **System of Linear Equations:**
   Solve the system:
   \[
   \begin{aligned}
   2x + y &= 5 \\
   3x - y &= 4
   \end{aligned}
   \]

**Solution:**
   Add both equations to eliminate \(y\):
\[
(2x + y) + (3x - y) = 5 + 4 \\
5x = 9 \\
x = \frac{9}{5}
\]

Substitute \(x = \frac{9}{5}\) into the first equation:
\[
2\left(\frac{9}{5}\right) + y = 5 \\
\frac{18}{5} + y = 5 \\
y = 5 - \frac{18}{5} = \frac{25}{5} - \frac{18}{5} = \frac{7}{5}
\]

Thus, the solution is:
\[
x = \frac{9}{5}, \quad y = \frac{7}{5}
\]

### 4. **Quadratic Equation:**
   Solve for \(x\):
   \[
   x^2 - 5x + 6 = 0
   \]

**Solution:**
   Factor the quadratic equation:
\[
x^2 - 5x + 6 = (x - 2)(x - 3) = 0
\]

Thus, the solutions are:
\[
x = 2 \quad \text{or} \quad x = 3
\]

### 5. **Inequality - Solve for \(x\):**
   \[
   2x - 7 \leq 3
   \]

**Solution:**
\[
2x - 7 \leq 3 \\
2x \leq 3 + 7 \\
2x \leq 10 \\
x \leq \frac{10}{2} \\
x \leq 5
\]

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### Do you want more details or have any specific questions about these solutions?

Here are 5 related exercises for practice:

1. Solve for \(x\): \(4x + 3 = 15\)
2. Solve for \(y\) in terms of \(x\): \(5x - 2y = 7\)
3. Solve the system: \(\begin{aligned} x + y &= 6 \\ 2x - y &= 3 \end{aligned}\)
4. Solve the quadratic equation: \(x^2 - 4x - 12 = 0\)
5. Solve the inequality: \(3x + 5 > 11\)

**Tip:** Always check your solutions by substituting them back into the original equations to verify their correctness!

Explore step-by-step solutions for algebraic exercises including linear equations, quadratic equations, systems of equations, and inequalities. Learn how to solve problems like 3x - 5 = 10, x^2 - 5x + 6 = 0, and more. Suitable for Grades 8-10.

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