🧮 algebra

1. **State the problem:** It takes 48 minutes for 7 people to paint 7 walls. We want to find how many minutes it takes 20 people to paint 20 walls. 2. **Understand the relationship

1. **State the problem:** It takes 48 minutes for 7 people to paint 7 walls. We want to find how many minutes it takes 20 people to paint 20 walls. 2. **Understand the relationship

1. **State the problem:** Solve the equation $ (4x-7)(x^2+2x) = 3 $ for $x$. 2. **Use the distributive property:** Expand the left side by multiplying each term in $4x-7$ by each t

1. **State the problem:** We need to compute the sum of the sequence: 3 + 6 + 9 + ... + 117 + 120 + 117 + ... + 9 + 6 + 3. 2. **Analyze the sequence:** This sequence increases by 3

1. The problem is to draw the graph of the function $f(x) = 4^x$. 2. The function $f(x) = a^x$ where $a > 0$ and $a \neq 1$ is an exponential function.

1. **State the problem:** Solve the linear equation $7x = 60 + 2x$ for $x$. 2. **Formula and rules:** To solve linear equations, we isolate the variable on one side by performing i

1. **State the problem:** Evaluate the expression inside the brackets and then write the function $y = x^3$ multiplied by that evaluated value. 2. **Evaluate the bracketed expressi

1. **Problem Statement:** Define even and odd functions, then find the domain and range of the function $f(x) = 1 + x^2$. 2. **Definitions:**

1. نبدأ بتحديد المشكلة: حل المعادلات الجبرية. 2. المعادلة هي تعبير رياضي يحتوي على مجهول (عادة x) وعلامة مساواة.

1. The problem is to understand the concept of a function in mathematics. 2. A function is a relation between a set of inputs and a set of possible outputs where each input is rela

1. Let's start by stating the problem: We want to understand how to solve a simple algebraic equation with the help of an example. 2. Consider the equation $2x + 3 = 7$. Our goal i

1. The problem: A magic square is a grid where the sums of numbers in each row, column, and diagonal are equal. 2. The formula: For an $n \times n$ magic square, the magic constant

1. **Problem:** Prove that for any natural number $k$, $1 \cdot k = k$. 2. **Formula and rules:** Multiplication by 1 is the identity operation in natural numbers, meaning multiply

1. **State the problem:** Prove by mathematical induction that $$\sum_{i=1}^n i^4 = \frac{n(n+1)(6n^3 + 9n^2 + n - 1)}{30}.$$\n\n2. **Base case (n=1):** Calculate the left side: $$

1. **State the problem:** We want to analyze the function $$y=\frac{1}{\sqrt{x^2+4x+3}}$$ and understand its behavior. 2. **Rewrite the expression inside the square root:** Factor

1. **Problem:** In an arithmetic progression (AP), the first term $a_1=2$ and the sum of the 1st and 6th terms is $16 \frac{1}{2} = 16.5$. Find the 4th term $a_4$. 2. **Formula and

1. **State the problem:** We have the equation $$3|x - 8| = k$$ where $$k$$ is a constant. We want to find the value(s) of $$\frac{k}{3}$$ such that the equation has exactly one so

1. The problem is to write the expression $4x3 \times \frac{2}{-3} - (-1) \times 3 \times (-3) \times 2$ in proper mathematical format. 2. First, interpret the expression carefully

1. Problem: Multiply $\frac{5y^3}{32x}$ and $\frac{-4}{15x^2 y^3}$ and simplify. Step 1: Multiply numerators and denominators:

1. **State the problem:** Simplify the expression $4 \times 3 \times \frac{2}{-3} - (-1) \times 3 \times (-3) \times 2$. 2. **Recall the rules:**

1. **State the problem:** Solve the equation $0.10x + 0.20y = 0.19(x + y)$ for $y$ in terms of $x$. 2. **Write the equation:**