🧮 algebra

1. **State the problem:** Solve the equation $$18 = \frac{a}{3} - 12$$ for the variable $a$. 2. **Add 12 to both sides** to isolate the term with $a$:

1. **State the problem:** Find the perpendicular distance from the point $P(5, -2)$ to the line given by the equation $$y = \frac{4}{3}x - 11.$$ 2. **Formula used:** The distance $

1. **Problemstellung:** Gegeben sind die Funktionen: $$f(x) = -2 \cdot 4^x, \quad g(x) = 1{,}5 \cdot 6^x, \quad h(x) = -4 \cdot 0{,}5^x, \quad k(x) = 4 \cdot 2^x$$

1. **State the problem:** Solve the equation $-36a^2 = 8a$ for $a$. 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero:

1. The problem asks to express $\left(\frac{y}{2}\right)^{-4}$ in the form $a y^n$ where $a$ and $n$ are integers. 2. Recall the rule for negative exponents: $x^{-m} = \frac{1}{x^m

1. The problem asks to express $\left(\frac{y}{2}\right)^{-4}$ in the form $a y^n$ where $a$ and $n$ are integers. 2. Recall the rule for negative exponents: $x^{-m} = \frac{1}{x^m

1. **Problem:** Simplify $x^4 x^{-2}$ with only positive exponents. 2. **Formula and rules:** When multiplying powers with the same base, add the exponents: $a^m \cdot a^n = a^{m+n

1. **Problem 7:** You save 90 every 2 weeks. How long to save 450? 2. Use the formula: Total saved = (Amount saved per period) \times (Number of periods).

1. **State the problem:** Simplify the expression $$\frac{(n-k)(n-1)! + k(n-1)!}{k!(n-k)!}$$. 2. **Factor the numerator:** Notice that both terms in the numerator share a common fa

1. Problem 8: Find the initial speed of a car that skids 108 feet using the formula $v = \sqrt{35 \times \text{skid length}}$. 2. Substitute the skid length 108 feet into the formu

1. **State the problem:** Solve the equation $$\frac{35}{x} = \frac{5}{2}$$ for $x$. 2. **Use the cross-multiplication method:** When two fractions are equal, their cross products

1. **State the problem:** Solve the equation $$\frac{(x+3)(x-2)}{(x+3)(x-1)} = \frac{x-1}{x-2}$$ for $x$.

1. **Stating the problem:** Solve the equation $$\frac{x}{6} = \frac{15}{18}$$ for $x$. 2. **Formula and rules:** To solve for $x$ in a proportion $\frac{a}{b} = \frac{c}{d}$, use

1. **State the problem:** Simplify the expression $$\frac{(x+3)(x-2)}{(x+3)(x-1)} - \frac{7-2}{x-1}$$ 2. **Rewrite the expression:** The second fraction numerator simplifies to $7-

1. Das Problem lautet: Schreibe die Dezimalzahl 0,875 als Bruchzahl. 2. Wir wissen, dass 0,875 eine Dezimalzahl mit drei Nachkommastellen ist, also können wir sie als \frac{875}{10

1. **State the problem:** We have the formula $$N = \frac{6000}{5p + z}$$ where $N$ is the number of radios sold, $p$ is the price per radio, and $z$ is a constant. We know that wh

1. The problem involves understanding and comparing percentages given: 95%, 85.5%, 18%, and 16%. 2. Percent means "per hundred," so 95% means 95 out of 100, or 0.95 as a decimal.

1. **State the problem:** Professor Leisurely takes twice as much time to correct 40 exams as Professor Swift takes to correct 60 exams. We want to find how many exams Swift can co

1. **Stating the problem:** We are given the formula for $N$ as $$N = \frac{6000}{5p + z}$$ where $p$ and $z$ are variables.

1. The problem is to understand and analyze the formula for $N$ given by $$N = \frac{6000}{5p + z}.$$\n\n2. This formula expresses $N$ as a function of two variables $p$ and $z$. T

1. **Stating the problem:** We want to find the valid formula for the value $y$ at time $t$ during the interval $0 \leq t \leq T$, given that $y$ starts at $y_s$ and decreases to $