🧮 algebra

1. The problem involves interpreting Y-shaped branching diagrams with labeled branches. 2. Each branch label represents a value or fraction associated with that branch.

1. The problem states the equation $65 = 98$. 2. This is a simple equality statement.

1. **State the problem:** We are given the rational function $$r(x) = \frac{4x^2}{x^2 - 2x - 3}$$ and we want to analyze its properties such as domain, intercepts, and asymptotes.

1. **Problem Statement:** Simplify the given expressions using index laws. 2. **Index Laws Used:**

1. Problem: Calculate the value of $\frac{8!}{2! \times 6!}$. Formula: $\binom{n}{r} = \frac{n!}{r!(n-r)!}$, here $\frac{8!}{2!6!}$ is a combination $\binom{8}{2}$.

1. **State the problem:** Simplify the expression $$\frac{2xz^2 + 7x^2z}{12xz^2 + 36x^2z - 120x^3} \div \frac{12z^2 + 42xz}{x^3z + 5x^4}$$. 2. **Rewrite division as multiplication

1. **State the problem:** Simplify the expression $$\frac{3n^{4} - 12n^{2}x^{2}}{8nx^{2} - 4n^{2}x} \cdot \frac{-8x}{n^{2} - 2nx - 8x^{2}}$$

1. **State the problem:** Evaluate the expression $3a + 2b - 6c$ when $a = 10$, $b = 9$, and $c = 4$. 2. **Write the expression:**

1. **State the problem:** Evaluate the expression $bc + 12.3$ when $a = 10$, $b = 9$, and $c = 4$. 2. **Identify the formula:** The expression is $bc + 12.3$.

1. **State the problem:** Evaluate the expression $a + 9c$ when $a = 10$, $b = 9$, and $c = 4$. 2. **Formula and explanation:** The expression is $a + 9c$. This means we take the v

1. **State the problem:** Evaluate the expression $a^2 - 18$ when $a = 10$, $b = 9$, and $c = 4$. Note that $b$ and $c$ are not used in this expression. 2. **Formula used:** The ex

1. **State the problem:** Solve the equation $-4.9y = 79.87$ for $y$. 2. **Formula and rules:** To isolate $y$, divide both sides of the equation by the coefficient of $y$, which i

1. Stating the problems: Evaluate the expression $$c^2 + 6$$ when $$a=10$$, $$b=9$$, and $$c=4$$.

1. **State the problem:** Solve the system of equations using Gaussian elimination: $$\begin{cases} 2x - y + 3z = 15 \\ x + 2y - 2z = -12 \\ 2x - 3y - 4z = 2 \end{cases}$$

1. **Problem Statement:** Define two functions $f: \mathbb{N} \to \mathbb{N}$ where \(\mathbb{N}\) is the set of natural numbers. (a) Find a function that is one-to-one (injective)

1. The problem is to understand the difference between an onto function and a one-to-one function. 2. A function $f: A \to B$ is called **onto** (surjective) if for every element $

1. **Problem Statement:** Define a function $f: \mathbb{N} \to \mathbb{N}$ that is one-to-one but not onto. 2. **Formula and Explanation:** Consider the function

1. Statement of the problem: Factor $x^2$. 2. Formula and important rules: For a positive integer $n$ we have $$x^n = x \cdot x \cdots \cdot x$$.

1. Problem (a): A businessman sold a refrigerator for 2745 making a profit of 15% on the cost price. We need to find the cost price (CP). 2. Formula: Profit% = \frac{Profit}{Cost P

1. **Problem 1:** A teacher's salary was 3300 after a 10% increase. Find the salary if the increase was 20% instead. 2. **Formula:** New Salary = Original Salary \times (1 + Increa

1. Problem 1: A teacher's salary was 3300 after a 10% increase. Find the salary if the increase was 20% instead. 2. Formula: New Salary = Original Salary \times (1 + Increase Rate)