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 :
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!Ask a new question for Free
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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
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