🧮 algebra

1. **Problem statement:** Find $2A - B$ where $A = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 13 & 6 \end{bmatrix}$ and $B = \begin{bmatrix} -2 & -3 \\ -5 & 1 \\ 4 & 2 \end{bmatrix}$.

1. We are asked to simplify the expression $\frac{999 \times 998}{999 \times 999} + 999$. 2. First, notice the fraction: $\frac{999 \times 998}{999 \times 999}$. We can cancel out

1. **Evaluate expressions using a 12-hour clock where ⊕ means addition modulo 12 and ⊖ means subtraction modulo 12.** - a. $3 \oplus 5 = (3 + 5) \mod 12 = 8$

1. The problem asks: Which parabola has its branches closest to the Oy-axis among the following options? 2. All given parabolas have the form $y=ax^2$. The distance of the parabola

1. Diberikan operasi biner $a@b = \frac{a}{b} + \frac{b}{a} - 2$. Kita akan menyelesaikan persamaan $$((c + 1)@c)@(c - 1) = (c + 1)@(c@(c - 1))$$

1. The problem states several equalities for a function $P(x)$: $P(0)=0$, $P(3)=a=P(5)$ and $P(11)=2a=P(13)$. 2. This means the function $P(x)$ takes the same value $a$ at $x=3$ an

1. Problem: Solve the quadratic equation $$x^2 - 5x + 6 = 0$$. 2. Step 1: Identify coefficients: $$a=1$$, $$b=-5$$, $$c=6$$.

1. Let's start by understanding the concept of a line equation. A line in a plane can be represented by the equation:

1. The equation of a line is given by $y = m \times x + c$, where $m$ is the slope and $c$ is the intercept. 2. The slope $m$ represents how steep the line is and is calculated as

1. Problem: Find the quotient and remainder when dividing the polynomial $$2x^3 + x^2 - x - 4$$ by $$x - 2$$. 2. Use polynomial long division:

1. El problema plantea analizar la función g(x) = x + 2 y la desigualdad \(|x+1|<3\). 2. La desigualdad \(|x+1|<3\) significa que la distancia entre x y -1 es menor que 3, es decir

1. **Problem statement:** Prove by mathematical induction that $$4^2 + 7^2 + 10^2 + \dots + (3n + 1)^2 = \frac{1}{2} n (6n^2 + 15n + 11)$$ for all positive integers $n$. 2. **Base

1. La línea recta que representa la ecuación $y = -3x + 2$ para los valores $x = 2$ y $x = 3$: Calculemos los valores de $y$:

1. Despejar $x$ en la ecuación $\sqrt{x} - 4 = w$. Para despejar $x$, primero sumamos 4 a ambos lados:

1. Despejar la variable $D$ en la ecuación $$A = \frac{Dd}{2}$$: Multiplicamos ambos lados por 2:

1. **State the problems:** 6. Prove the geometric series sum formula: $$a + ar + ar^2 + \cdots + ar^{n-1} = a \frac{1 - r^n}{1 - r}$$

1. The given expression is \( \frac{1-r^k}{1-r} \).\n2. This expression is a common formula in algebra called the sum of a geometric series.\n3. It represents the sum of the series

1. The problem is to simplify the expression $$\frac{1-r^{k-1}}{1-r}$$. 2. Recognize that the denominator is a difference of terms involving $r$, commonly appearing as the denomina

1. Given the identity to prove: $(n \text{ choose } r) + (n \text{ choose } r-1) = (n+1 \text{ choose } r)$. This is a key binomial coefficient identity. 2. Recall the definition o

1. The problem statement "solve for rhs before lhs" is unclear because typically in equations we solve for a variable rather than for "rhs" or "lhs" which stand for right-hand side

1. **Prove by induction that** $a + ar + \cdots + ar^{n-1} = a \frac{1-r^n}{1-r}$ for $r \neq 1$. Step 1: Base case ($n=1$):