🧮 algebra

1. **Problem statement:** Solve the system of linear equations $$\begin{cases} x - 2y + z = 1 \\ 2x + y - z = 4 \\ 3x - y - 4z = m \end{cases}$$

1. **Problem statement:** We have a traffic network with nodes A, B, C, D and variables $x_1, x_2, x_3, x_4, x_5$ representing vehicle flows on certain roads. Given known flows and

1. **Problem a:** Describe the transformations from $y=f(x)$ to $y=f(3x)+2$. 2. The transformation $y=f(3x)$ compresses the graph horizontally by a factor of $\frac{1}{3}$ because

1. **State the problem:** We are given the function $$f(x) = \frac{x^2 + 4ax + 6a^2}{(x + 2a)(x + 3a)}$$ where $a$ is a positive constant. We need to express $f(x)$ in partial frac

1. **Problem (a):** Given that $f(x) = (x - a)^2 g(x)$, where $f(x)$ and $g(x)$ are polynomials, show that $(x - a)$ is a factor of $f'(x)$. 2. **Step 1:** Differentiate $f(x)$ usi

1. **Problem Statement:** Evaluate exactly the sum $$\sum_{i=25}^{150} \left(\frac{1}{i+4} - \frac{1}{i+5}\right)$$

1. **State the problem:** Solve the equation $$3 \times 2^{x+1} = 4 \times 3^{2x-3}$$ for $x$, and give the answer correct to 3 significant figures. 2. **Rewrite the equation:**

1. **Problem Statement:** (a) Sketch the graph of $y = |x + 3a|$, where $a$ is a positive constant.

1. **State the problem:** Since the user did not specify a particular problem, let's consider a common algebra problem: Solve the quadratic equation $$ax^2 + bx + c = 0$$ for $x$.

1. **State the problem:** Factor the quadratic expression $x^2 + 5x + 6$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multipl

1. The problem asks us to find the value of the function $f(x) = \frac{4.92}{x}$ at $x = 2.46$. 2. The formula to use is simply substitution: $f(2.46) = \frac{4.92}{2.46}$.

1. The problem asks us to find the value of the function $f(x) = x^{0.8}$ when $x = 7$. 2. The function rule is $f(x) = x^{0.8}$, which means we raise $x$ to the power of $0.8$.

1. The problem asks us to find the value of the function $f(x) = -5.14 - 2.5x$ at $x = -3.88$. 2. The formula to evaluate the function at a given $x$ is to substitute the value of

1. The problem asks us to find the value of the function $f(x) = \frac{8.2}{x} + 10.2$ when $x = 1$. 2. The formula given is $f(x) = \frac{8.2}{x} + 10.2$. To find $f(1)$, substitu

1. **Stating the problem:** We are given two expressions:

1. **Problem statement:** A man buys an article for 10% less than its value and sells it for 10% more than its value. We need to find his gain or loss percentage. 2. **Let the valu

1. **State the problem:** We need to put the fractions $\frac{4}{9}$, $\frac{5}{12}$, and $\frac{2}{5}$ in ascending order. 2. **Formula and rules:** To compare fractions, convert

1. **Problem Statement:** Solve the equation or inequality involving the variable $\omega$ given by $(\omega - 1) \neq \omega, \omega + 1$ and the expressions $(\omega) q r \omega

1. **بيان المسألة:** لدينا المتتالية التراجعية المعرفة بالعلاقة: $$U_0 = 1, \quad U_{n+1} = \frac{4U_n}{U_n + 2}, \quad n \in \mathbb{N}$$

1. لنبدأ بفهم المشكلة: لدينا مجموعتان من الأعداد الصحيحة، الأولى هي $\{3,4,5,7,8\}$ والثانية هي $\{1,2,3,4,5,6,7,8\}$. المطلوب هو فهم العلاقة بين هذه المجموعات أو التعبير عنها بطري

1. **State the problem:** Find the point-slope form of the equation of a line passing through the point $(2,-1)$ with a negative slope, given the graph information. 2. **Recall the