🧮 algebra

1. **State the problem:** Find the equation of a line in slope-intercept form $y = mx + b$ where the gradient (slope) $m = -3$ and the line passes through the point $(4, 2)$. 2. **

1. The problem is to evaluate the expression $1 - 5x^2 + 5x + 5$ for $x = 1$. 2. Substitute $x = 1$ into the expression:

1. Problem 1: Given $f(x) = \frac{(1+2x)^2}{(1-x)^2}$ (i) Find the first 4 terms in the power series expansion.

1. **Problem:** Solve the equation $2x + 3 = 7$. 2. **Formula and Trick:** To solve linear equations like this, use the trick "undo operations in reverse order". Start by undoing a

1. **Problem Statement:** You want a quick introduction and tricks to solve problems on Geometric Progression (GP) for your exam. 2. **Introduction to GP:** A Geometric Progression

1. **State the problem:** Solve the equation $x^6 + x^6 = 2^{32}$ for $x$. 2. **Combine like terms:** Since both terms are $x^6$, we can add them:

1. The problem is about solving an equation and understanding the operations allowed on both sides. 2. When solving equations, the key rule is to perform the same operation on both

1. **State the problem:** Solve the linear equation $5a + 2 = -13$ for $a$. 2. **Recall the formula and rules:** A linear equation in one variable can be solved by isolating the va

1. Masalah yang diberikan adalah memahami rumus $f(n) + f(n+1) + f(n+2) + \ldots + f(2n)$ dan mengapa bentuk penjumlahan seperti itu muncul. 2. Rumus ini adalah penjumlahan dari fu

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: $$