🧮 algebra

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

1. **Problem:** Determine if the function $f(x) = x^{-1} + x^7 + x^{-3}$ is even, odd, or neither. 2. **Recall definitions:**

1. The letter $t$ in math often represents a variable, commonly used to denote time in equations and functions. 2. It is a placeholder that can take on different values depending o

1. The problem states that the uppercase letter $R$ is 34 and asks to transfer it to the lowercase letter $r$ which is 17. 2. This suggests a relationship or conversion between $R$

1. The problem is to analyze the sign (positive, negative, or zero) of six different graphs or functions. 2. To determine the sign of a function, we look at the values of the funct

1. **Stating the problem:** Simplify the expression $x\sqrt{x}$. 2. **Recall the rule:** The square root of $x$ is $\sqrt{x} = x^{\frac{1}{2}}$.

1. **State the problem:** We have two pipes filling a swimming pool. The larger pipe runs for 4 hours, the smaller pipe for 9 hours, and together they fill half the pool. The small

1. **Determine whether each equation is linear in the given variables.** Linear equations in variables $x_1, x_2, x_3$ or $x, y$ have the form $$a_1 x_1 + a_2 x_2 + a_3 x_3 = b$$ o

1. **State the problem:** Solve the inequality $\frac{x+3}{2} \geq 5$. 2. **Formula and rules:** To solve inequalities involving fractions, multiply both sides by the denominator (

1. **State the problem:** Solve the inequality $$\frac{x}{9} - 6 > 4$$. 2. **Isolate the variable term:** Add 6 to both sides to get rid of the -6.

1. **Problem statement:** A number is first increased by 25% and then decreased by 20%. We need to find the net percentage change. 2. **Formula and explanation:**