🧮 algebra

1. **Stating the problem:** We are given several fractions and mixed numbers: $\frac{1}{4}$, $7 \frac{1}{5}$, $2 \frac{1}{8}$, $\frac{2}{3}$, and $\frac{4}{9}$. The problem involve

1. **Problem:** Rewrite the line $3x - 4y + 12 = 0$ in slope-intercept form, identify slope and y-intercept, and convert to intercept form. 2. **Formula:** Slope-intercept form is

1. **Problem Statement:** Verify that two lines are perpendicular by using their slopes. 2. **Formula and Rule:** Two lines are perpendicular if and only if the product of their sl

1. **State the problem:** We are given a line $L$ with equation $5x - y = 8$ and a point $(1, -3)$. We need to find the equation of the line perpendicular to $L$ that passes throug

1. Simplify the following numerical expressions step-by-step. (a) Simplify $2 + 3 - 5(6 + 7 \times 3)$:

1. **Problem:** Solve the system using Cramer's rule: $$\begin{cases} 3x + 2y = 7 \\ 4x - y = 2 \end{cases}$$

1. **State the problem:** Solve the compound inequality $$-1 < x + 4 < 3$$ and determine which number line correctly represents the solution. 2. **Isolate the variable:** To solve

1. **State the problem:** We are given a line $L$ with equation $5x - y = 8$ and a point $(1, -3)$. We need to find the equation of the line perpendicular to $L$ that passes throug

1. **State the problem:** Find all integer solutions $x$ such that $$12 < 3(x + 2) \leq 27.$$\n\n2. **Write the inequality:** $$12 < 3(x + 2) \leq 27.$$\n\n3. **Divide all parts of

1. **Problem Statement:** Find the equation of a line perpendicular to a given line. 2. **Formula and Rules:**

1. Prime Factorization and Number Theory 1.1) Express 660 as a product of its prime factors.

1. The problem is to solve the compound inequality $$14 \leq 6x + 4 \leq 24$$ and express the solution as a fraction in simplest form. 2. We use the property that if $$a \leq b \le

1. **State the problem:** Solve the compound inequality $$-2 < 2x + 11 < 12$$ for $x$. 2. **Understand the inequality:** This is a double inequality, meaning $2x + 11$ is greater t

1. **Stating the problems:** - Find $k$ if $f(3k - 2) = 2k + 3$ given $f(x) = 4x + 1$.

1. The problem asks which equation represents an inverse variation. Inverse variation means $y$ varies inversely as $x$, so $y = \frac{k}{x}$.

1. **Problem Statement:** Convert improper fractions to mixed numbers and mixed numbers to improper fractions. 2. **Formulas and Rules:**

1. **Calculate the area of quadrilateral ABCD** Since the problem does not provide coordinates or side lengths, we cannot calculate the area without additional information.

1. **Evaluate** $\frac{2 + \sqrt{2}}{\sqrt{2}}$. Use the property $\frac{a+b}{c} = \frac{a}{c} + \frac{b}{c}$.

1. Problem: A machine fills 680 bottles in 5 hours. How many bottles will it fill in 3 hours? Formula: Bottles filled are directly proportional to time.

1. **State the problem:** Solve the proportion $4 : x = 9 : 18$ to find the value of $x$. 2. **Recall the rule for proportions:** If $a : b = c : d$, then $\frac{a}{b} = \frac{c}{d

1. **State the problem:** We want to simplify and understand the expression $$63.72 + 1.96 \left( \frac{1.9}{\sqrt{n}} \right)$$ where $n$ is a positive number.