🧮 algebra

1. **Problem statement:** The sum of two numbers is 18 and the sum of their reciprocals is \(\frac{9}{40}\). Find the numbers. 2. **Formulate the equations:** Let the two numbers b

1. **State the problem:** Convert the equation $2x - 4y + 11 = 0$ into normal form. 2. **Recall the normal form of a line:** The normal form is given by $$x \cos \alpha + y \sin \a

1. **State the problem:** Solve the linear equation $9x + 11 = 3$ for $x$. 2. **Isolate the variable term:** Subtract 11 from both sides to get the term with $x$ alone.

1. Let's state the problem: What happens when a number is raised to the power of 0? 2. The rule for exponents states that for any nonzero number $a$, $a^0 = 1$.

1. **Проблем:** Намерете корените на квадратно уравнение с коефициенти $a=0$, $b=0$, $c=1$. 2. **Формула:** Квадратното уравнение е от вида $$ax^2 + bx + c = 0.$$

1. The problem is to simplify $3^4 \times 3^2$ into a single expression using indices. 2. The rule for multiplying powers with the same base is: $$a^m \times a^n = a^{m+n}$$ where

1. The problem is understanding how the fraction $-\frac{3}{2}$ can become positive. 2. The sign of a fraction depends on the signs of the numerator and denominator.

1. **State the problem:** Solve the system of equations using substitution method: $$-2x + 3y = 0$$

1. The problem is to find the sum of the fractions $\frac{1}{3}$, $\frac{1}{4}$, and $\frac{1}{6}$. 2. To add fractions, we need a common denominator. The denominators are 3, 4, an

1. **State the problem:** Solve the system of equations using substitution method: $$2x + y = 7$$

1. সমস্যাটি হলো: একটি ভগ্নাংশের LCM নির্ণয় করা, যেখানে ভগ্নাংশগুলোকে সমান হরফে রূপান্তর করতে হবে। 2. LCM (Least Common Multiple) বা সর্বনিম্ন সাধারণ গুণক নির্ণয়ের জন্য প্রথমে ভগ্

1. **הבעיה:** יוסי רץ במסלול שבו בדקה הראשונה הוא רץ 75 מטרים, ובכל דקה הבאה המרחק גדל פי 1.02 מהמרחק שברץ בדקה הקודמת. 2. **חישוב המרחק שרץ יוסי בדקה הרביעית:**

1. **State the problem:** Simplify the expression $$\frac{1}{2} - \frac{3}{2} + \frac{2}{3}$$. 2. **Find a common denominator:** The denominators are 2, 2, and 3. The least common

1. **Stating the problem:** We want to understand how the cost per person changes when different numbers of people stay in the villa. 2. **Given information:** The villa has a fixe

1. **State the problem:** We need to perform polynomial division of $$4x^4 - 5x^3 + x^2$$ by $$3x^2 - 4x + 1$$. 2. **Recall the division process:** Polynomial division is similar t

1. **State the problem:** We have a polynomial $$p(x) = 10x^3 + ax^2 - 10x + b$$ where $$a$$ and $$b$$ are integers.

1. The problem is to solve the equation $$\frac{1}{2} \div \frac{3}{2} + 2 \frac{1}{2} = 3$$. 2. First, recall that dividing by a fraction is the same as multiplying by its recipro

1. The problem is to simplify the fraction $\frac{11}{95}$. 2. The formula for simplifying a fraction is to divide the numerator and denominator by their greatest common divisor (G

1. The problem is to solve the system of equations given by the user. 2. Since the user did not specify the exact equations, I will explain the general approach to solving a system

1. **Stating the problem:** We are asked to perform matrix multiplications and solve for unknown variables in given matrix equations. 2. **Matrix multiplication formula:** For matr

1. **Simplify each radical expression:** 2. **Problem 1:** Simplify $\sqrt{32} - 2\sqrt{72} + \sqrt{200}$.