🧮 algebra

1. **State the problem:** Find the points of intersection between the line defined by the system $$\begin{cases} 2x - y + 2z - 12 = 0 \\ 2x - 4y - z + 6 = 0 \end{cases}$$

1. **State the problem:** Solve the equation $$9x^4 - 82x^2 + 9 = 0$$ for $x$. 2. **Use substitution:** Let $y = x^2$. Then the equation becomes a quadratic in $y$:

1. Problemi: Rishkruani shprehjet sipas udhëzimeve dhe thjeshtoni shprehjet algjebrike. 2. Formula dhe rregulla të rëndësishme: Në algjebër, shprehjet mund të thjeshtohen duke grup

1. **Problem:** Make $a$ the subject of the formula $2a + b = 10$. 2. **Formula and rules:** To isolate $a$, subtract $b$ from both sides, then divide both sides by 2.

1. **State the problem:** Solve the equation $$x^6 + 7x^3 - 8 = 0$$ for $x$. 2. **Use substitution:** Let $y = x^3$. Then the equation becomes $$y^2 + 7y - 8 = 0$$ because $x^6 = (

1. **State the problem:** Find the parametric equations of the line of intersection of the two planes given by: $$2x - y + 2z - 12 = 0$$

1. 题目是询问是否可以使用截距式来解决问题。 2. 截距式通常用于直线方程，形式为 $\frac{x}{a} + \frac{y}{b} = 1$，其中 $a$ 和 $b$ 分别是 $x$ 轴和 $y$ 轴的截距。

1. 题目要求求双曲线 $\frac{x^2}{4} - \frac{y^2}{5} = 1$ 的焦点坐标。 2. 双曲线标准形式为 $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$，其中焦点坐标为 $(\pm c, 0)$，且满足关系 $$c^2 = a^2 + b^2$$。

1. **State the problem:** We are given a rectangle with area 36 cm². The length is labeled as $x$ and the width as $x - 5$. We need to find the dimensions of the rectangle. 2. **Fo

1. نبدأ بحل التمرين الثاني وهو حساب قيمة الدالة $$g(x) = \sqrt{x} + \frac{2}{|x-1|} - 3$$ عند $$x=9$$. 2. نعوض $$x=9$$ في الدالة:

1. **بيان المسألة:** لدينا الدوال:

1. **Problem 1.1.1: Solve the quadratic equation** $$x^2 - 4x + 3 = 0$$ 2. **Formula:** For quadratic equations $$ax^2 + bx + c = 0$$, solutions are given by the quadratic formula:

1. **Problem:** Solve the equation $$x^2 - 4x + 3 = 0$$ 2. **Formula:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=1$$, $$b=-4$$, and $$c=3$$.

1. **State the problem:** Solve the equation $$\frac{7}{10}v + 1 = \frac{83}{15} - v$$ for $v$. 2. **Write down the equation:** $$\frac{7}{10}v + 1 = \frac{83}{15} - v$$

1. **Problem Statement:** Given data points for $x$ and $y$, fit a quadratic model $y = b_0 + b_1 x + b_2 x^2$ using the least squares method.

1. **State the problem:** A store buys a skateboard for 70 and applies a 15% markup. We need to find the selling price. 2. **Formula:** Selling Price = Cost Price + Markup

1. **State the problem:** We have a backpack originally priced at 30 and it is discounted by 40%. We need to find the sale price.

1. **Problem:** How many triangular tiles are there in the FIFTH figure of the sequence? 2. **Understanding the pattern:** The problem states the sequence of triangular tiles incre

1. **Problem:** How many triangular tiles are there in the FIFTH figure of the sequence? 2. **Understanding the pattern:** The sequence shows triangular tiles increasing in a patte

1. **Problem:** Given that $9x^{2} + 6xy + 4y^{2}$ is a factor of $27x^{3} - 8y^{3}$, find the other factor. 2. **Step 1: Recognize the form of the expression.**

1. **State the problem:** We have 7 numbers with a mean of 4. Six of the numbers are 2, 3, 1, 2, 7, and 8. We need to find: a) The seventh number.