🧮 algebra

1. **State the problem:** We need to find a fraction $\frac{x}{y}$ such that: - When the numerator is multiplied by 4 and the denominator is decreased by 2, the fraction becomes $\

1. Problem: Find the value of $f(1/x) + \frac{1}{f(x)}$ where $f(x) = \frac{1-x}{1+x}$. Formula: Substitute $1/x$ into $f$ and simplify.

1. The problem is to simplify $3\sqrt{-81}$.\n\n2. Recall that the square root of a negative number involves the imaginary unit $i$, where $i=\sqrt{-1}$.\n\n3. We can rewrite $\sqr

1. The problem is to analyze the linear function given by the equation $y = -\frac{1}{2}x + 3$. 2. This is a linear equation in slope-intercept form $y = mx + b$, where $m$ is the

1. **Problem Statement:** Solve the system of linear equations: $$\begin{cases} 3x + 3y + 2z = 1 \\ x + 2y = 4 \\ 10y + 3z = -2 \\ 2x - 3y - z = 5 \end{cases}$$

1. **State the problem:** Find the sum of the first 60 terms of the geometric sequence \(\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \ldots\). 2. **Identify the first term and common r

1. **State the problem:** Solve the equation $-2 + y = -7$ for $y$. 2. **Formula and rules:** To isolate $y$, add 2 to both sides of the equation to maintain equality.

1. **State the problem:** Solve the equation $6 + \frac{y}{3} = 2(1 + y)$ for $y$. 2. **Write the equation:**

1. **State the problem:** Solve the system of equations by substitution: $$4.51w = 13.14 - 2.33v$$

1. مسئله: یافتن مقدار $a - b + c - d$ برای تابع وارون $f^{-1}(x) = ax + b|x + c| + d$ که وارون تابع $f(x) = 3x + |x + 1|$ است. 2. ابتدا تابع $f(x)$ را بررسی می‌کنیم:

1. Muammo: $f(x) = 2x^2$ va $\varphi(x) = x + 1$ bo'lsa, $f(\varphi(x)) = \varphi(f(x))$ tenglama nechta $x$ qiymatida to'g'ri kelishini toping. 2. Formulalar:

1. Masala: $y = x^2$ va $y = |x|$ funksiyalarining $OX$ o'qida yotmaydigan kesishish nuqtalari orasidagi masofani toping. 2. Kesishish nuqtalarini topish uchun tenglama tuzamiz: $$

1. Muammo: $f(x) = \frac{x^2 - 9}{x + 3}$ funksiyaning $x \to 3$ da limitini toping. 2. Formulalar va qoidalar: Limitni topishda, agar ifoda $\frac{0}{0}$ ko'rinishida bo'lsa, ifod

1. **State the problem:** Solve the system of linear equations by substitution: $$4.51w = 13.14 - 2.33v$$

1. The problem asks to rewrite the expression $\sqrt{-62}$ as a complex number using the imaginary unit $i$. 2. Recall that the imaginary unit $i$ is defined as $i = \sqrt{-1}$.

1. The problem is to rewrite the expression $\sqrt{-24}$ as a complex number and simplify all radicals. 2. Recall that the imaginary unit $i$ is defined as $i = \sqrt{-1}$.

1. The problem is to simplify the expression $12 - \sqrt{-13}$.\n\n2. Recall that the square root of a negative number involves imaginary numbers. Specifically, $\sqrt{-a} = i\sqrt

1. The problem is to rewrite the expression $\sqrt{-39}$ as a complex number and simplify all radicals. 2. Recall that the imaginary unit $i$ is defined as $i = \sqrt{-1}$.

1. **Stating the problem:** Simplify the expression $$\frac{2m^2 - 2m - 24}{2m + 6} \div \frac{2m^2 - 32}{3m + 12}$$. 2. **Rewrite the division as multiplication by the reciprocal:

1. The problem asks us to rewrite the expression $20 - \sqrt{-19}$ as a complex number using the imaginary unit $i$. 2. Recall that the imaginary unit $i$ is defined as $i = \sqrt{

1. **State the problem:** Rewrite the expression $\sqrt{-98}$ as a complex number and simplify all radicals. 2. **Recall the formula and rules:** For any negative number under a sq