🧮 algebra

1. **Stating the problem:** Simplify and evaluate the expression $$\left(\frac{2\sqrt{2} - \sqrt{5}}{\sqrt{3}}\right)^{2023} \times \left(\frac{2\sqrt{2} + \sqrt{5}}{\sqrt{3}}\righ

1. Given the matrix equation: $$\begin{pmatrix} 3 & 2 \\ 7 & -2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 12 \\ 8 \end{pmatrix}$$

1. The problem is to divide one polynomial by another using long division. 2. Start by dividing the leading term of the dividend by the leading term of the divisor to find the firs

1. The problem states that the $n^{th}$ term of a sequence is given by $$a_n=(-2)^{\frac{n}{2}}.$$ 2. We are asked to find the $2n^{th}$ term of the sequence, denoted as $a_{2n}$.

1. **Problem (a):** A hire tool firm's net return decreases by 10% annually. The current net gain is 400. Find the total of all future profits assuming the tool lasts forever. 2. S

1. Solve (a) $\frac{n}{3} x - 7 + 5 \le 6$. Simplify the left side: $\frac{n}{3} x - 2 \le 6$.

1. **Find the features of line A**. Line A passes through points (1,0) and (0,6).

1. **State the problem:** Simplify the expression $$\frac{xy}{x^2 - y^2} - \frac{y}{x + y}.$$\n\n2. **Factor the denominator:** Notice that $$x^2 - y^2$$ is a difference of squares

1. **Find the gradient between each pair of points given:** Gradient formula between points $(x_1,y_1)$ and $(x_2,y_2)$ is

1. Stating the problem: Simplify the expression $$\frac{2a + 5}{a + b} \times \frac{a - b}{xy} - \frac{y}{x^2 - y^2} \div \frac{x + y}{1}$$

1. The problem is to understand how the value of $m$ in linear equations of type $y = mx + c$ affects the graph. 2. We have two linear functions given:

1. Stating the problem: A car covers 180 kilometers in 3 hours. We want to find how many kilometers it covers in 5 hours. 2. Determine the speed of the car by dividing the distance

1. Let's first clarify the problem: we need to determine the equation of line $l_2$. 2. Usually, the equation of a line can be expressed as $y=mx+b$, where $m$ is the slope and $b$

1. The problem states we have a line $l_1$ with equation $x + 2y = 4$. 2. We need to find line $l_2$ perpendicular to $l_1$ and passing through the origin.

1. تمّ إعطاء السؤال: حساب حاصل ضرب 480 في 5. 2. نستخدم قاعدة الضرب المباشر للأعداد الصحيحة: 480 \times 5.

1. **Stating the problem:** We need to factor common terms from the expression $$y'=(4x+3)^3(x+1) - 4\left[16(x+1) - 3(4x+3)\right].$$

1. Stating the problem: Three businessmen Njoroge, Chatles, and Omondi contributed a total of 329000 to start a business. We know the ratio of Njoroge's to Chatles's contribution i

1. Let's understand what a common factor is: it is a number or expression that divides two or more terms without leaving a remainder. 2. To find a common factor in step 6, identify

1. **State the problem:** Solve for $x$ in the equation $$\frac{125^{4x}}{5^{2x}} = 3125.$$\n\n2. **Express all terms with the same base:** Note that $125 = 5^3$ and $3125 = 5^5$.

1. نبدأ بفهم المطلوب: 300 قسمة خمس تعني 300 ÷ 5. 2. نكتب العملية الحسابية:

1. The problem asks how to find the unit rate given a fraction or ratio in the form $\frac{\text{numerator}}{\text{denominator}}$. 2. The unit rate is the value corresponding to "p