Math Problem Statement

Set related exercises with solution

Solution

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 xx:

3x5=103x - 5 = 10

Solution:

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$$ --- ### Do you want more details or have any specific questions about these solutions? Here are 5 related exercises for practice: 1. Solve for $$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!

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Systems of Equations
Quadratic Equations
Inequalities

Formulas

Linear equation: ax + b = c
Quadratic equation: ax^2 + bx + c = 0
System of linear equations: solving by substitution or elimination
Inequalities: ax + b ≤ c

Theorems

Addition property of equality
Factorization method for quadratic equations
Elimination method in systems of equations

Suitable Grade Level

Grades 8-10