🧮 algebra

1. **Problem 1: Solve the system by graphing and analyze the number of solutions** Given system:

1. **State the problem:** We have a kite-shaped quadrilateral with sides labeled as follows: top side = $3y$, left side = $5x - 15$, right side = $2x + 3$, and bottom side = $6y -

1. **State the problem:** We need to graph the system of linear inequalities: $$y > -3x + 2$$

1. **Problem:** Find $n(P)$ where $P = \{x: 3 < x \leq 11\}$ and $x$ is an integer. **Step 1:** Identify integers satisfying $3 < x \leq 11$.

1. **Problem Statement:** Solve the equation $2x^2 - 4x - 6 = 0$. 2. **Formula Used:** For quadratic equations of the form $ax^2 + bx + c = 0$, the solutions are given by the quadr

1. **Problem:** Find $n(P)$ where $P = \{x: 3 < x \leq 11\}$ and $x$ is an integer. Step 1: Identify integers satisfying $3 < x \leq 11$.

1. **Problem statement:** We have two functions:

1. The problem involves finding missing numbers in a set of rectangular boxes where arrows indicate multiplication or division relationships between the center number and the numbe

1. **State the problem:** We need to divide 240 into two parts in the ratio 2:3 and find the smaller share. 2. **Formula and explanation:** When a quantity is divided in the ratio

1. **Stating the problem:** We need to divide 240 into two parts in a given ratio and find the smaller share. 2. **Understanding the ratio:** Suppose the ratio is $a:b$. The total

1. Let's consider a simple algebra problem: Solve for $x$ in the equation $2x + 3 = 7$. 2. The formula to isolate $x$ is to perform inverse operations to both sides of the equation

1. **State the problem:** Given $a=5$ and $b=3$, solve the equation $$ (2a)^2 = 3b $$ for the values of $a$ and $b$. 2. **Write the formula and substitute values:** The equation is

1. The problem is to graph the system of inequalities: $$\begin{cases} 2x - 5y > 3 \\ 3x - 7y \leq 4 \\ 5x + 3y \geq 1 \\ x - 6y \geq 2 \end{cases}$$

1. We are given a system of inequalities: $$\begin{cases} 2x - 5y > 3 \\ 3x - 7y \leq 4 \\ 5x + 3y \geq 1 \\ x - 6y \geq 2 \end{cases}$$

1. The problem is to graph the relationship between variables $x$ and $y$. 2. To graph $y$ as a function of $x$, we need a specific equation or relationship between $x$ and $y$. Si

1. The problem is to graph the inequality $2x - 5y > 3$. 2. Rewrite the inequality in slope-intercept form $y < \frac{2}{5}x - \frac{3}{5}$.

1. **Problem 30:** A traveler goes 8 km upstream and then returns downstream. The stream speed is 4 km/h, and the upstream trip took 20 minutes longer than the downstream trip. Fin

1. **State the problem:** Simplify the expression $1+1-1+4@-1$. 2. **Identify the operations:** The expression contains addition (+), subtraction (-), and an unknown symbol '@'. Si

1. State the problem: Compute the sum $\sum_{i=1}^{n} i^2$ for positive integer $n$.\n2. Goal and formula: We will prove the closed form\n$$\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}

1. The problem is to understand and simplify the expression $\frac{a}{b}$. 2. This is a fraction where $a$ is the numerator and $b$ is the denominator.

1. **State the problem:** Solve the equation $4r + 30 = 2$ for $r$. 2. **Write the formula and rules:** This is a linear equation. To solve for $r$, isolate $r$ by performing inver