🧮 algebra

1. State the problem: Solve and simplify the equation $$16x^2+4y^2+32x-16y-32=0$$. 2. Group terms by variables: $16x^2+32x + 4y^2 -16y = 32$

1. We are given the equation of a conic: $$16x^2 + 25y^2 + 160x + 200y + 400 = 0$$. 2. First, group the $x$ and $y$ terms:

1. **State the problem:** We are asked to graph the inequality $-3x + 2y > 6$ and determine where the shaded region is. 2. **Rewrite the inequality:** To graph the inequality, firs

1. **State the problem:** We need to graph the inequality $3x - y \leq 4$ and find three points that satisfy this condition. 2. **Rewrite the inequality:** Rearrange to a more fami

1. Problem: Graph the inequality $$3x - y \leq 4$$ and show the shaded region. 2. Rewrite the inequality to express $$y$$ in terms of $$x$$:

1. The problem is to graph the inequality $3x - y \leq 4$. 2. First, rewrite the inequality to express $y$ in terms of $x$:

1. **Problem Statement:** Find the ratio in which the point $(a,3)$ divides the line segment joining the points $(11,-2)$ and $(3,6)$, and calculate the value of $a$. 2. **Step 1:

1. Let's start by identifying the formula you provided in your first task. 2. Once we have the formula, we can break it down into its components and understand its structure.

1. The problem: Solve $2x+3=7$ for $x$. 2. Subtract $3$ from both sides: $2x=7-3$

1. **State the problem:** Simplify the expression $$\frac{5^{x - y} \times 125^{3x - y}}{25^x}$$. 2. **Rewrite bases with the same base if possible:** Note that 125 and 25 can be e

1. **State the problem:** Solve the given exponential and logarithmic expressions and simplify the fraction involving powers of 5 and 25. 2. Evaluate the expression $x \log 6 = 7 \

1. The problem is to solve the equation $X^2 - 8X - 5 = (X - 4)^2 - 11$ for $X$. 2. First, expand the right side: $(X - 4)^2 = X^2 - 8X + 16$.

1. Stating the problem: Solve the equation $9x - 10 = 5x + 2(2x - 5)$ for $x$. 2. Expand the right side: $2(2x - 5) = 4x - 10$.

1. State the problem: Solve the equation $9x - 10 = 5x + 2(2x - 5)$. 2. Distribute the 2 on the right side: $9x - 10 = 5x + 4x - 10$.

1. The problem is to clarify the meaning of "nsqrt" notation. 2. In this notation, "n" is not a multiplier but rather the index or indicator of the root.

1. Let's begin by stating the problem: how to factorize algebraic expressions. 2. Factorization means expressing an algebraic expression as a product of its factors.

1. **Stating the problem:** Simplify the expression $$\frac{(4^{n+1} + 4^n)^2}{(2^{n+1} - 2^n)^2}$$. 2. **Rewrite bases:** Note that $$4 = 2^2,$$ so rewrite powers of 4 in terms of

1. (a) Given matrices: $$A = \begin{pmatrix}2 & 4 \\ 1 & 3\end{pmatrix},\quad B = \begin{pmatrix}3 & 2 \\ 5 & 1\end{pmatrix}$$

1. Stated problem: Solve the equation $$\frac{4^{2}+1+4^{2}}{2^{2}+1-2^{2}\cdot 2}=25$$. 2. Simplify powers and expressions in numerator and denominator:

1. We want to solve the inequality $x < \sqrt{x^2 + 1}$.\n\n2. Notice that the square root expression $\sqrt{x^2 + 1}$ is always positive because $x^2 + 1 \geq 1 > 0$ for all real

1. The problem shows two expressions with fractions and variables $t$ and $x$ nearby. The expressions are: $$\frac{3+1}{1-1}x$$