🧮 algebra

1. **Stating the problem:** Solve the system of equations: $$\begin{cases} x + 3y - z = -2 \\ 3x + 6y - 3z = -5 \\ 2x + 4y + z = \frac{11}{3} \end{cases}$$

1. Planteamos el problema: En una progresión geométrica (PG), la suma de los 6 primeros términos es 28 veces la suma de los 3 primeros términos. Debemos encontrar la razón $r$ de l

1. Planteamos el problema: Tenemos una progresión aritmética (P.A.) con 23 términos y el término central es 15. Queremos calcular la suma de todos los términos. 2. Recordemos que e

1. **State the problem:** Solve the rational equation $$\frac{9}{x^2 - 6x + 9} = \frac{3x}{x^2 - 3x}$$ and state any restrictions on $x$. 2. **Identify restrictions:** The denomina

1. Wir sollen die Gleichung $ (x + 24) \cdot (x - 24) = 0 $ lösen. 2. Die Nullregel besagt: Ein Produkt ist genau dann null, wenn mindestens ein Faktor null ist.

1. **Planteamiento del problema:** Resolver la expresión a) $7 - [-5 + (2 - 9 + 13) - 25]$ paso a paso. 2. **Fórmulas y reglas importantes:** Para resolver expresiones con paréntes

1. The problem is to simplify the expression $-3x^2 - 2x + 1$. 2. This is a quadratic expression in standard form $ax^2 + bx + c$ where $a = -3$, $b = -2$, and $c = 1$.

1. **State the problem:** We need to analyze the function $$y = -4x^5 + \sqrt{x^3} + 3x^2 - 7$$ and understand its components. 2. **Rewrite the function:** Recall that $$\sqrt{x^3}

1. **הבעיה:** נתונה הפונקציה $$f(x) = x^3 + 6x^2 + 9x$$. יש לקבוע עבור אילו ערכים של $$k$$ המשוואה $$f(x) = k$$ יש שני פתרונות. 2. **הבנת הבעיה:** המשוואה $$f(x) = k$$ שווה למציאת

1. **State the problem:** Solve the simultaneous equations:

1. **State the problem:** Find $x$ if $9x^2 + 3x + 1 = 28$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:

1. **Evaluate the expression:** $$\frac{-12 \times (-3) \times 4 - (-20)}{-6 \times 6 + 4 + (-6)}$$

1. **State the problem:** Solve the equation $$\log_2 \left( \log_2 (7x - 10) \times \log_x 16 \right) = 3$$ and find the sum of all solutions. 2. **Recall logarithm properties:**

1. **State the problem:** Rearrange the equation $$\frac{p}{q r} = 3 + \frac{1}{q}$$ to make $r$ the subject. 2. **Start with the given equation:**

1. The problem is to prove that points lie on a line using the gradient (slope). 2. The formula for the gradient between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $$m = \frac{y_2

1. Énoncé du problème : Résoudre l'équation logarithmique $\log_2 (x + 1) = \log_2 5$. 2. Formule utilisée : Si $\log_a A = \log_a B$ alors $A = B$ (à condition que $a > 0$, $a \ne

1. **State the problem:** We want to simplify and understand the function $$y = \left(4 \ln(x^3)\right)^{\frac{1}{4}}$$. 2. **Recall the logarithm rule:** $$\ln(x^3) = 3 \ln(x)$$.

1. সমস্যাটি হলো: $a=3^{\frac{1}{3}}+3^{-\frac{1}{3}}$ হলে, $\frac{a^6-1}{a^3}$ এর মান নির্ণয় করতে হবে। 2. প্রথমে $a$ এর মানকে সহজভাবে প্রকাশ করি। এখানে $a = x + \frac{1}{x}$ যেখান

1. **State the problem:** Simplify the expression $$(b - 3b^2 + 7)(2 - 3b)(5 + b)$$. 2. **Recall the distributive property:** To multiply multiple polynomials, multiply two at a ti

1. **State the problem:** We want to find the least number of coins using 1¢, 5¢, 10¢, and 25¢ coins to make 143 cents (which is $1.43). 2. **Formula and approach:** Use the greedy

1. The problem is to generalize the equation $h = -5(t-2)^2 + 20$. 2. This equation is in vertex form of a quadratic function: $$h = a(t - h)^2 + k$$ where $(h, k)$ is the vertex a