🧮 algebra

1. **State the problem:** Solve the system of equations: $$2x - y = 3 + 5x^2$$

1. **State the problem:** Find the equation of the line passing through the points (10, 9) and (5, 12) in slope-intercept form $y = mx + b$. 2. **Find the slope $m$ using the slope

1. **Problem:** Write an equivalent expression using the commutative property for $y + 9$. 2. **Commutative Property:** This property states that the order of addition or multiplic

1. **State the problem:** Find the inverse function $f^{-1}(x)$ of the function $f(x) = \sqrt{x} + 5$. 2. **Recall the definition of inverse function:** The inverse function $f^{-1

1. Enunciamo il problema: risolvere il sistema di equazioni $$\begin{cases} (x - 2)^2 + \frac{y + 4}{3} = (x - 5)(x + 1) + 2y - x \\ \frac{x}{5} + \frac{y}{3} = 2 \end{cases}$$

1. Problema: Risolvi il sistema lineare $$\begin{cases}(3x - y)^2 - 2x + 3 = (3x - y)(3x + y) + 2y^2 - 6xy + y \\ \frac{x}{3} - \frac{y}{2} = 1\end{cases}$$

1. Multiply each expression by distributing the factor outside the parentheses. 2. For problem 1: Multiply 5 by each term inside the parentheses:

1. The problem is to find the discriminant of a quadratic equation and determine how many solutions it has. 2. The discriminant $\Delta$ of a quadratic equation $ax^2 + bx + c = 0$

1. **State the problem:** Solve the quadratic equation $2x^2 - 4x + 5 = 0$. 2. **Recall the quadratic formula:** For an equation $ax^2 + bx + c = 0$, the solutions are given by

1. The problem is to divide 3 by the fraction \(\frac{17}{5}\). 2. Recall the rule for dividing by a fraction: \(a \div \frac{b}{c} = a \times \frac{c}{b}\).

1. Aufgabe: Löse die Gleichung und überprüfe die Lösung mit der Probe. Gegeben: $$2x + 3 = 11$$

1. The problem asks us to fill in the blanks in the equation $$10 + 35 = 5 \times (\_ + \_)$$ so that both sides are equal. 2. First, simplify the left side:

1. Das Problem lautet: Berechne den Ausdruck für $\sum_{i=1}^n i$ ohne die Variable $i$ einzeln auszuwerten. 2. Die Formel für die Summe der ersten $n$ natürlichen Zahlen ist:

1. **State the problem:** Solve the linear equation $8 + 9x = 53$ for $x$. 2. **Isolate the variable term:** Subtract 8 from both sides to get the term with $x$ alone on one side.

1. **Problemstellung:** Löse die Gleichung $-4 \cdot i = 24$ nach $i$ auf und mache die Probe. 2. **Formel und Regeln:** Um $i$ zu isolieren, teile beide Seiten der Gleichung durch

1. The problem is to solve the equation $12 = f + 7$ for $f$. 2. To isolate $f$, subtract 7 from both sides of the equation.

1. **State the problem:** Solve the equation $\frac{x}{2} = 6$ for $x$. 2. **Recall the formula and rule:** To isolate $x$, multiply both sides of the equation by 2, because multip

1. **State the problem:** Solve the linear equation $$17 = 5x + 2$$ for $x$. 2. **Isolate the variable term:** Subtract 2 from both sides to get the term with $x$ alone on one side

1. **State the problem:** We need to find the value of $p$ in the equation $$2 + p = 7$$. 2. **Use the formula:** To isolate $p$, subtract 2 from both sides of the equation.

1. **State the problem:** Solve the equation $10a = 60$ for $a$. 2. **Formula and rules:** To solve for $a$, we need to isolate $a$ on one side of the equation. Since $a$ is multip

1. **State the problem:** Solve the equation $9 = x$ for $x$. 2. **Formula and rules:** This is a simple equation where the variable $x$ is already isolated on one side.