🧮 algebra

1. Сформулюємо умову задачі: вектори $\mathbf{a} = (5, 3, x)$ та $\mathbf{b} = (y, 6, 4)$ перпендикулярні, тобто їх скалярний добуток дорівнює нулю: $$\mathbf{a} \cdot \mathbf{b} =

1. **State the problem:** Simplify the expression $$\frac{1}{2} + \frac{3}{4} - \frac{1}{3} \times \frac{2}{5} \div \frac{4}{7} + \frac{2}{6}$$. 2. **Handle multiplication and divi

1. **State the problem:** Simplify the expression $$\sqrt{\frac{4x}{10}} \div \sqrt{\frac{10}{9x}}$$. 2. **Rewrite the division of square roots as a single square root:**

1. **State the problem:** We are given the system of equations: $$3x^2 + 2xy + 4y + 2y^2 = 4$$

1. **Problem 8:** Alvin and Simon share 540 in ratio 4:5. Alvin's share = $\frac{4}{4+5} \times 540 = \frac{4}{9} \times 540 = 240$. Simon’s share = $\frac{5}{9} \times 540 = 300$.

1. **State the problem:** A company sells a product at 1200 per unit and charges a delivery fee of 200 per product. If the total order value plus delivery fees exceeds 5000, a 4% d

1. Problem 4: Carly and James share money in ratio 5:3. Carly gets 70 more than James. Find James' amount. 2. Let James' amount be $x$. Then Carly's amount is $x + 70$.

1. **Problem:** Molly, Paige, and Demi share 42 sweets in the ratio 3 : 2 : 1. Find how many sweets each receives. 2. **Step 1:** Add the parts of the ratio: $3 + 2 + 1 = 6$ parts

1. **Question 2: Molly, Paige and Demi share 42 sweets in the ratio 3 : 2 : 1.** Step 1: Add the parts of the ratio: $3 + 2 + 1 = 6$ parts total.

1. **Problem 2:** Molly, Paige, and Demi share 42 sweets in the ratio 3 : 2 : 1. 2. First, find the total parts of the ratio: $3 + 2 + 1 = 6$ parts.

1. **State the problem:** Determine whether the equation $x - 4 = -4 + x$ has one solution, no solutions, or infinitely many solutions. 2. **Simplify both sides:** The left side is

1. Stating the problem: Solve the equation $$\frac{x}{9} = 6$$ for $x$. 2. To isolate $x$, multiply both sides of the equation by 9 to cancel the denominator:

1. Stating the problem: Solve for $x$ in the equation $$10 = \frac{4}{5} (7x + 15)$$. 2. Eliminate the fraction by multiplying both sides by 5:

1. The problem states that a cooking workshop started at 10:00 and ended at 13:00. 2. We need to find how many sessions of length $\frac{3}{4}$ hours fit into the total duration.

1. The original point is given as $(8, 2)$. This means $x=8$ and $y=2$ where $y=f(x)$. 2. The transformation is $y = -3f(-4x)$. We need to find the new coordinates after applying t

1. The problem is to simplify the expression given by the user. 2. Since no specific expression was provided, please provide the expression you want to simplify.

1. **State the problem:** Simplify and verify the equality: $$1 - \frac{1}{\sqrt{8}} = \left(\frac{1}{4} + \frac{1}{\sqrt{8}}\right) - 1$$

1. We are asked to find between which two consecutive integers the value of $x$ lies when $2^x = 14$. 2. To solve for $x$, take the logarithm base 2 of both sides:

1. \textbf{הבעיה:} יש לנו שלוש טבלאות עם פרמטרים $a_k$, $b_k$, ו-$c_k$ עבור $k=1,2,3$. הנוסחה הכללית עבור הערך במקום $(m,n)$ בטבלה $k$ היא: $$a_k + c_k (m-1) + \left(b_k + 30(m-1)\

1. The problem is to simplify the expression $8y^2 - 4y$. 2. Identify the common factor in both terms. Both terms have a factor of $4y$.

1. The problem is to simplify the expression $\frac{8y^2 - 4y}{4y}$.\n\n2. First, factor the numerator: $8y^2 - 4y = 4y(2y - 1)$.\n\n3. Rewrite the expression using the factorizati