🧮 algebra

1. **Problem statement:** We want to find the value(s) of $k$ such that the equation $f(x) = k$ has exactly one solution. 2. **Understanding the problem:** The equation $f(x) = k$

1. **Problem:** Solve the system of equations using matrix method: $$\begin{cases} x + 2y = 7 \\ x + 3y + 4z = 16 \\ x + y + z = 7 \end{cases}$$

1. Bài toán yêu cầu giải phương trình hoặc hệ phương trình (vui lòng cung cấp phương trình cụ thể để giải). 2. Công thức và quy tắc giải phương trình thường bao gồm:

1. Bài toán đầu tiên là giải phương trình $$\frac{x}{15} = -\frac{4}{5}$$. 2. Ta sử dụng quy tắc nhân chéo để giải phương trình dạng phân số: $$x \times 5 = -4 \times 15$$.

1. Énoncé du problème : Trouver la valeur de l'expression a) $\log\left(\left(\log_4 49\right)^3\right) \log_9^2 + \log\left(\frac{1}{0^2}\right) \sqrt{2}$. 2. Analyse de l'express

1. **State the problem:** We have an arithmetic progression (AP) where the first term $a_1 = 300$ metres and the common difference $d = 50$ metres. The total number of rounds is 10

1. **Problem Statement:** A closed cuboid made of paper has a volume of 150 cm³. The height is $h$, the width is $w$, and the length is triple the width, i.e., $3w$.

1. The problem is to understand the concepts of linearity and continuity in graphs. 2. Linearity means the graph of a function is a straight line, which can be represented by the e

1. **Stating the problem:** We want to solve a simple algebraic example: Solve for $x$ in the equation $2x + 3 = 7$. 2. **Formula and rules:** To solve for $x$, we use the rule of

1. The problem is to understand the notation for solution sets using intervals and unions. 2. The notation $s=]-\infty,-value]\cup[-value,value]\cup[value,\infty[$ represents the u

1. **State the problem:** Solve the quadratic equation $2x^2 + 8x = 960$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:

1. The problem is to solve the equation given (though no explicit equation was provided, we assume a general approach to solving algebraic equations). 2. To solve an algebraic equa

1. ปัญหาข้อ 25: \(2^{x+y} = 6^{y}\) และ \(3^{x-1} = 2^{y+1}\) 2. เริ่มจากเขียนสมการทั้งสองในรูปที่ใช้ฐานเดียวกันหรือแปลงให้ง่ายขึ้น

1. **State the problem:** We need to find the 10th term of a geometric sequence where the first term $a_1 = -5$ and the common ratio $r = -2$. 2. **Formula for the $n^{th}$ term of

1. **State the problem:** We need to find the sum of all terms in a geometric series where the first term $a_1=4$ and the second term $a_2=-3$. 2. **Recall the formula for the sum

1. **State the problem:** Find the 10th term of a geometric sequence where the 1st term $a_1 = -3$ and the 4th term $a_4 = 81$. 2. **Recall the formula for the $n$th term of a geom

1. **State the problem:** Calculate the sum of the first 6 terms of a geometric series with first term $a_1 = -21$ and common ratio $r = -\frac{2}{3}$. 2. **Formula for sum of firs

1. **State the problem:** We need to find the sum of the first 12 terms of a geometric series where the first term $a_1 = -3$ and the second term $a_2 = 6$. 2. **Find the common ra

1. The problem is to convert mixed numbers to improper fractions. 2. The formula to convert a mixed number $a \frac{b}{c}$ to an improper fraction is:

1. **State the problem:** We have a geometric sequence where the first term $a_1 = 6$ and the second term $a_2 = 18$. We need to find the 8th term $a_8$. 2. **Recall the formula fo

1. **Problem:** Given vectors $a = \left(-7, -15\right)$ and $b = \left(10, 16\right)$, find $8\left(a + \frac{1}{2}b\right)$. 2. **Formula and rules:** To solve this, we use vecto