🧮 algebra

1. **Problem statement:** We want to find the minimum value of $$S = \frac{12 x^3}{y^2} + \frac{y}{x^2} + 3y$$

1. The problem asks for the possible values of $x$ and $y$, but no specific equation or condition is given. 2. To find possible values of variables, we need an equation or inequali

1. **State the problem:** Simplify the expression $$\frac{\sqrt{24x^3}}{\sqrt{3x^2}} - \sqrt{50x}$$. 2. **Use the property of radicals:** $$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{

1. **State the problem:** Solve the linear equation $$3309x + 6442y = 13682$$ for one variable in terms of the other. 2. **Formula and rules:** This is a linear equation in two var

Given the user's request to remove numbering and explanations, here is the solution only: Solve the equation $2x + 3 = 7$.

1. **State the problem:** We are given values $a = \sqrt{81}$, $b = -\frac{3}{4}$, and $c = \pi$. We need to determine which set represents the irrational numbers among the options

1. The problem is to express the decimal number 0.5 as a fraction. 2. Recall that decimals can be converted to fractions by considering the place value of the decimal digits. For 0

1. The problem is to convert a decimal or number into a fraction. 2. To convert a decimal to a fraction, write the decimal number as the numerator (top number) and use a power of 1

1. **State the problem:** We are given that quantities $x$ and $y$ are proportional, and we have pairs of values: $(9, 4.5)$, $(14, 7)$, and $(30, 15)$. We need to find the constan

1. **Stating the problem:** Solve the equation $$\frac{x-5}{3} = \frac{x-3}{3} + \frac{6x+1}{21}$$ for $x$. 2. **Formula and rules:** To solve equations with fractions, find a comm

1. **Problem statement:** Solve the equation $$\frac{x-5}{3} - \frac{x-3}{3} = \frac{6x+1}{21}$$. 2. **Formula and rules:** To solve equations with fractions, first find a common d

1. **Problem statement:** We have several sequences $(u_n)$ defined by different formulas and recurrence relations. We need to analyze their properties such as bounds, monotonicity

1. The problem is to solve the equation $8^z = -7$ for $z$. 2. Recall that for any real number base $a > 0$ and $a \neq 1$, the exponential function $a^x$ is always positive. This

1. **State the problem:** Convert and calculate various time-related quantities. 2. **Find the number in 4 1/2 hours:**

1. The problem is to evaluate the expression $-5 + 8$. 2. The operation involved is addition of integers.

1. **Problem Statement:** Determine all intervals where the function $f(x) \leq 0$ based on the given graph description. 2. **Understanding the problem:** The function $f(x)$ is le

1. **State the problem:** We have a shape ABCDEF made from two rectangles with given side lengths in terms of $x$. The total area is 342 cm$^2$. We need to show that $x^2 + x - 72

1. The problem asks us to determine if the equation $4x + 2y - 10 = 0$ represents a straight line and explain why. 2. The general form of a linear equation in two variables $x$ and

1. **Problem statement:** Find the equation of the straight line passing through the points $(1,4)$ and $(-2,-2)$ in the form $y = mx + c$. 2. **Formula used:** The slope $m$ of a

1. Problem: Find the value of the function $f(x) = 2x^2 - 3x + 1$ at $x=3$. 2. Formula: To find the value of a function at a specific point, substitute the value of $x$ into the fu

1. **State the problem:** Rewrite the equation $$\frac{1}{2}x - \frac{3}{4}y = \frac{5}{6}$$ in the form $$y = mx + c$$ and find the gradient $$m$$. 2. **Formula and rules:** The s