🧮 algebra

1. The problem is to verify if a given answer satisfies an equation or condition by plugging it in. 2. To check correctness, substitute the proposed answer into the original equati

1. **State the problem:** Solve the exponential equation $$9^{3x+4} = 27^{4x+3}$$ for $x$. 2. **Recall the formula and rules:** Both 9 and 27 can be expressed as powers of 3 since

1. **State the problem:** Evaluate $$\sqrt[3]{\frac{0.119 \times 0.256}{0.068} \times 7}$$ without using a calculator or table, leaving the answer as a simplified fraction. 2. **Re

1. **State the problem:** Simplify the expression $$(z-(3+i))(z-(3-i))$$. 2. **Recall the formula:** This is a product of two binomials of the form $(z - a)(z - \overline{a})$ wher

1. The problem asks: Which number equals $\log_a a$? 2. Recall the definition of logarithms: $\log_a b = c$ means $a^c = b$.

1. **Stating the problem:** We need to solve questions 3 and 4 using direct and indirect proportion (ratio and proportion). 2. **Understanding direct and indirect proportion:**

1. **Problem 1: Find the mean, median, and mode of the numbers:** 17, 18, 16, 17, 17, 14, 22, 15, 16, 17, 14, 12. 2. **Mean** is the average of the numbers. Formula: $$\text{Mean}

1. The problem is to understand what "10 QUESTION" means in a math context. Since it is ambiguous, we interpret it as a request to solve or explain a math problem involving the num

1. **Problem Statement:** Find the 7th Fibonacci number using the Fibonacci sequence formula. 2. **Formula:** The Fibonacci sequence is defined by the recurrence relation:

1. **State the problem:** Simplify the expression $$3a^2 + 4b + a + 5a^2 + 3b + 2a$$. 2. **Identify like terms:** Terms with the same variable and exponent can be combined.

1. Let's start by stating the problem: solving an algebra question generally means finding the value(s) of the variable(s) that satisfy the given equation or inequality. 2. The mos

1. The problem is to square each given value and then add all the squared values together. 2. The formula used is the sum of squares: $$\sum_{i=1}^n x_i^2$$ where $x_i$ are the giv

1. **State the problem:** Solve the system of equations using the Gauss-Jordan method: $$\begin{cases}-x + y + 2z = 2 \\ 3x - y + z = 6 \\ -x + 3y + 4z = 4\end{cases}$$

1. **State the problem:** Solve the quadratic equation $x^2 - 6x + 8 = 0$. 2. **Recall the quadratic formula:** For any quadratic equation $ax^2 + bx + c = 0$, the solutions are gi

1. Let's start with the ellipse. An ellipse is the set of all points where the sum of the distances from two fixed points (foci) is constant. 2. The standard form of an ellipse cen

1. **Problem statement:** Given points $P(-4,2)$ and $Q(5,-4)$, a line $l$ is drawn through $P$ and perpendicular to the line segment $PQ$. This line $l$ meets the $y$-axis at poin

1. **Problem Statement:** We are given the 8th term ($T_8$) and the 17th term ($T_{17}$) of a Harmonic Progression (HP) as $\frac{2}{5}$ and $\frac{2}{11}$ respectively. We need to

1. **Problem Statement:** We need to find and indicate the solution region for the system of linear inequalities: $$3x + 7y \leq 21$$

1. Stating the problem: We need to find the simplified form of the quotient of two expressions: $$\frac{x^2 - 5x}{x + 2} \div \frac{3x - 15}{x^2 - 4}$$

1. The problem is to find the sum of the first $n$ terms of the series $1\cdot 2 + 2\cdot 3 + 3\cdot 4 + \cdots$. 2. Each term in the series can be written as $k(k+1)$ where $k$ is

1. **State the problem:** We are given the 11th term ($a_{11}$) and 19th term ($a_{19}$) of an arithmetic progression (AP) as 42 and 74 respectively. We need to find the sum of the