🧮 algebra

1. **Stating the problem:** Consideriamo gli insiemi:

1. **Problem:** The Sultan of Brunei noticed that the ratio of emeralds to rubies is the same as the ratio of diamonds to pearls. Given 85 emeralds, 119 rubies, and 45 diamonds, fi

1. **State the problem:** Solve the system of equations using the elimination method: $$4x - 2y = 14$$

1. **Problemstellung:** Zeichne den Graphen der proportionalen Funktion $f$ mithilfe eines geeigneten Steigungsdreiecks für die Funktion $f(x) = 1,2x$. 2. **Formel:** Die Steigung

1. **Problem statement:** We have the function $f(x) = x^3 - 2x^2 - 4x + 8$. We need to calculate $f(x)$ for $x = -3, -1, 1, 4$, show that $f(x)$ can be factored as $f(x) = (x^2 -

1. **State the problem:** We know the Amazon rainforest decreased by 20% over 50 years, and its area after 50 years is 3,290,125 km^2.

1. **Stating the problem:** Vi har funktionen $f(x) = x^3$ och en tangent till grafen i punkten där $x = a$. Tangenten, den positiva x-axeln och linjen $x = a$ bildar en triangel.

1. **State the problem:** Solve the system of linear equations using the elimination method: $$5x + 4y = -14$$

1. The first problem asks to find the integer function value $f(-4)$ from the graph of the function $y=f(x)$. 2. The graph is a downward-opening parabola passing through points nea

1. **Problem Statement:** We are given a line segment from point $(-9,4)$ to point $(9,-6)$ on the Cartesian plane. We need to find the "rise" and "run" of the line and then calcul

1. **State the problem:** Find the slope of the line passing through the points $(-9, -10)$ and $(7, 7)$. 2. **Formula for slope:** The slope $m$ of a line through points $(x_1, y_

1. **Problem statement:** Der Wildbestand eines Naturparks nimmt exponentiell ab und sinkt innerhalb von 5 Jahren um 4%. Im Jahr 2010 wurden 1780 Tiere gezählt.

1. **Problem statement:** (a) Fully factorise $a^2 + 2ab + b^2 - 49$.

1. **State the problem:** We need to graph the inequality $y > \frac{1}{2}x - 6$. 2. **Understand the boundary line:** The boundary line is given by the equation $y = \frac{1}{2}x

1. **Stel het probleem vast:** We moeten de ongelijkheid $$-4 \cdot (-x - \frac{5}{4}) > x + 8$$ grafisch oplossen en de oplossingsverzameling aanvullen. 2. **Schrijf de ongelijkhe

1. The problem involves two fractions being added and then the third fraction multiplied by the result of that addition. 2. Let's denote the first two fractions as $\frac{a}{b}$ an

1. **Stating the problem:** Solve the equation $$\sqrt[10]{\frac{x^2 + \frac{x}{y}}{}} = \sqrt[10]{\left(x + \frac{1}{x^2}\right)^5} \cdot \sqrt{x}$$ given in exercise 123, page 69

1. The problem involves understanding and comparing different functions: a linear function $y = ax + b$, a reciprocal function $y = \frac{1}{x}$, and a quadratic function described

1. **State the problem:** Solve the logarithmic equation $$\log_6(-4x) - \log_6 7 = \log_6 54$$ for $x$. 2. **Recall the logarithm subtraction rule:** $$\log_b A - \log_b B = \log_

1. **State the problem:** Solve the equation $$\frac{2x - 1}{3} - \frac{3x + 1}{5} = 1$$. 2. **Identify the formula and rules:** To solve equations with fractions, find a common de

1. Stating the problem: Simplify the expression $$\frac{3x - 4}{5} + \frac{5x - 2}{5}$$. 2. Since both terms have the same denominator 5, we can combine the numerators over the com