🧮 algebra

1. **State the problem:** Solve the equation $16x - x - 8x = -14$ for $x$. 2. **Combine like terms:** On the left side, combine the terms with $x$:

1. **State the problem:** Solve the system of linear equations: $$3x - 2y = 38$$

1. **State the problem:** We need to graph the function $$f(x) = (x-1)^2 - 1$$. 2. **Recall the formula:** This is a quadratic function in vertex form $$f(x) = a(x-h)^2 + k$$ where

1. **State the problem:** Solve the system of linear equations: $$\begin{cases} x + 2y = 2 \\ x - y = 8 \end{cases}$$

1. **Problem a:** Solve the system by substitution: $$\begin{cases} y = 2x - 4 \\ y = 2x + 2 \end{cases}$$

1. **State the problem:** Solve the system of linear equations for part (a): $$6x + 12y = -6$$

1. The problem is to simplify the expression $16^{-\frac{3}{4}}$. 2. Recall the rule for negative exponents: $a^{-b} = \frac{1}{a^b}$.

1. 题目说明：已知函数 $$f(x) = \log_a \left( \frac{1 + kx}{x - 1} \right)$$，其中 $$a > 0, a \neq 1$$，且 $$f(x)$$ 是奇函数。求实数 $$k$$ 的值，并判断 $$f(x)$$ 在区间 $$(1, +\infty)$$ 上的单调性。 2. 奇函数定义：函数 $$f(x)$$

1. 题目说明：已知幂函数 $$f(x) = (m^2 - 2m - 2) x^{m-1}$$ 在区间 $$[0, +\infty)$$ 上单调递增，求： (1) 参数 $$m$$ 的值；

1. **问题陈述**：已知幂函数 $$f(x) = (m^2 - 2m - 2)x^{m-1}$$ 在区间 $$[0, +\infty)$$ 上单调递增，求 $$m$$ 的值。 2. **公式与规则**：

1. Planteamos el problema: Resolver la ecuación $3(x+1)=2(2x-3)$ para encontrar el valor de $x$. 2. Aplicamos la propiedad distributiva para eliminar los paréntesis:

1. Planteamos el problema: Resolver la ecuación $4x - 19 = 8 - 5x$ para encontrar el valor de $x$. 2. Usamos la propiedad de igualdad que dice que podemos sumar o restar el mismo v

1. **Problem statement:** Prove by mathematical induction that $n < 2^n$ for all $n \in \mathbb{N}$. 2. **Base case:** For $n=1$, check if $1 < 2^1$.

1. **State the problem:** Given the equations $a(x + y) = 7$ and $x - y = 3$, find the value of $8xy(x + y)x^2$. 2. **Analyze the given equations:**

1. **State the problem:** Given the equations $a(x + y) = 7$ and $x - y = 3$, find the value of $8xy(x + y)$. 2. **Identify knowns and unknowns:** We know $a(x + y) = 7$ and $x - y

1. The problem is to verify the values of $a$ and $b$ given as $a=65$ and $b=115$. 2. To check if these values are correct, we need the original equations or conditions involving $

1. **State the problem:** Solve the equation $$3 \log_x 4 + 2 \log_4 x 4 + 3 \log_{16} x \cdot 4 = 0$$ for real numbers $x$. 2. **Rewrite the logarithms:** Recall that $\log_a b =

1. Bài toán yêu cầu tìm giá trị của biểu thức $$A = (x + y - 4)^{2024} - y^{2024}$$ với phần X chưa rõ ràng, ta sẽ giải phần biểu thức chính trước. 2. Ta có biểu thức $$A = (x + y

1. **Problem statement:** A manufacturer wants to minimize the cost of producing a cylindrical can with volume 340 cm³.

1. **State the problem:** Find the equation of an ellipse given the center $(1,3)$, one focus $(1,0)$, and one vertex $(1,-1)$. 2. **Recall the standard form of an ellipse equation

1. The problem is to find the equation of the line passing through the points given in the link (assuming points are (2,3) and (4,7) for demonstration). 2. The formula for the slop