🧮 algebra

1. **State the problem:** We want to find the conditions on $a$ and $b$ for the system of linear equations: $$\begin{cases} 2x - 3y = a \\ 4x - 6y = b \end{cases}$$

1. **State the problem:** We want to determine if the inequality $\frac{1}{2} \geq \frac{e}{2}$ is true. 2. **Recall the inequality rule:** When both sides of an inequality are div

1. **Problem 6:** Find the total amount of money raised by the sixth, seventh, and eighth grades. 2. We know:

1. **State the problem:** Solve the quadratic equation $x^2 - x - 20 = 0$ by factorization. 2. **Recall the factorization method:** For a quadratic equation $ax^2 + bx + c = 0$, we

1. **Problem 6:** The sixth, seventh, and eighth grades raised money for their middle school. The sixth grade raised 40% of the total, the seventh grade raised 22%, and the eighth

1. **Problem Statement:** We are given two whole numbers $c$ and $d$. We need to determine which of the following statements about the parity (odd or even) of $2c + 3d$ are true: -

1. **State the problem:** We need to find the value of the expression $3a + 4b$ when $a = 5$ and $b = 8$. 2. **Formula and explanation:** The expression is a linear combination of

1. Problem 6: Find the total amount of money raised by the three groups (adults, parents, and eighth-grade class). Given:

1. The problem is not specified clearly, so I will explain how to approach solving a general algebraic problem. 2. Typically, to solve an algebraic equation, you isolate the variab

1. **Problem 4:** Toby has raised 225 which is 60% of his goal. We need to find how much more money Toby needs to raise to meet his goal. 2. **Formula:** To find the total goal amo

1. **Problem 4:** Toby has raised 225 which is 60% of his goal. We need to find how much more he needs to raise to meet his goal. 2. **Formula:** If $x$ is the goal amount, then $6

1. **Problem Statement:** Factor one quadratic factor of the polynomial $$x^6 + 20x^3 - 8$$ from the given options. 2. **Rewrite the polynomial:** Let $$y = x^3$$, then the polynom

1. Problem 16: Simplify $$\frac{a^2}{a-4} + \frac{8a - 16}{4 - a}$$. 2. Notice that $$4 - a = -(a - 4)$$, so rewrite the second fraction:

1. সমস্যাটি হলো: যদি $x^2 = 3x - 1$ হয়, তাহলে $x^3 + \frac{1}{2} x^2$ এর মান কত? এবং $x^6 - \frac{1}{x^3}$ এর মান নির্ণয় করতে হবে, যেখানে $x^4 = 47 - 2x^2$। 2. প্রথমে, $x^3$ এর ম

1. সমস্যাটি হলো একটি অ্যালজেব্রিক সমীকরণ সমাধান করা। 2. প্রথমে সমীকরণটি লিখুন এবং সমাধানের জন্য প্রয়োজনীয় সূত্র ব্যবহার করুন।

1. **Problem Statement:** We are given three expressions and asked to analyze or prove certain properties:

1. প্রথম প্রশ্ন: সমীকরণ $x^2 - 5x - 1 = 0$ যেখানে $x > 0$ এবং $a^2 + b^2 = c^2$। 2. (ক) উৎপাদকের বিভাজন করো: $x^2 + y^2 - 2xy - 1$

1. **Problem statement:** Given $x^2 = 3x - 1$, find: (a) $(x + \frac{1}{x})^2$

1. The problem involves solving a system of linear equations and understanding a 2x2 matrix. 2. The system of equations is:

1. **State the problem:** Find the domain of the function $$f(x) = \ln\left(\frac{1}{16^x} - 1\right)$$. 2. **Recall the domain rule for logarithms:** The argument of the natural l

1. The problem is to determine whether a system is convenient or not, and whether it is trivial or non-trivial. 2. In mathematics, especially in algebra and system theory, a system