🧮 algebra

1. **State the problem:** Solve the equation $$x + y = 9$$ for one variable in terms of the other. 2. **Formula and rules:** This is a linear equation in two variables. To express

1. The problem is to evaluate an algebraic expression or solve an algebraic problem. 2. Algebra involves using variables, constants, and operations like addition, subtraction, mult

1. **State the problem:** Find the slope of the line passing through points $(2, 0)$ and $(-14, 8)$. 2. **Formula for slope:** The slope $m$ of a line through points $(x_1, y_1)$ a

1. **Problem 7:** Calculate $\left(1 - \frac{3}{4}\right)\left(3 - \frac{1}{2}\right)$. 2. Simplify inside the parentheses:

1. **State the problem:** We need to find the selling price of a set of hand tools that cost $13.46$ with a $30\%$ markup. 2. **Formula used:** Selling Price = Cost + Markup

1. **State the problem:** Find the point of intersection (POI) of the system of equations: $$\text{(1) } x + 2y = 4$$

1. **State the problem:** We have a collection of eight integers where the mean, median, unique mode, and range are all equal to $8$. We want to find the largest integer in this co

1. **State the problem:** Convert the equation $$8x + 6y = -6$$ into slope-intercept form $$y = mx + b$$, where $$m$$ is the slope and $$b$$ is the y-intercept. 2. **Isolate $$y$$:

1. Problema 54: Risolvere $$\frac{1}{x^{2} - 2x + 1} + \frac{2}{x^{2} - 3x + 2} = - \frac{2}{2 - x}$$ 2. Problema 56: Risolvere $$\frac{2x^{2}}{2x - 1} = \frac{1}{4x - 2}$$

1. Problema 54: Risolvere l'equazione $$\frac{1}{x^2 - 2x + 1} + \frac{2}{x^2 - 3x + 2} = - \frac{2}{2 - x}$$ 2. Notiamo che $$x^2 - 2x + 1 = (x-1)^2$$ e $$x^2 - 3x + 2 = (x-1)(x-2

1. **State the problem:** Jill and Brian together invested $1900 in two accounts. Jill's account earns 4% interest, Brian's earns 3%, and the total interest earned is $70. We need

1. **Problem Statement:** We are given that $f(1) = f(-5)$ and want to determine if this implies the $h$ value of the vertex of the parabola is $-3$. 2. **Recall the vertex form of

1. **State the problem:** We have a quadratic function $$f(x) = ax^2 + 12x + c$$ with constants $$a$$ and $$c$$. The parabola opens upward, so $$a > 0$$. The vertex is at $$(h,k)$$

1. **State the problem:** Polly’s mother packed party bags with 4 sweets and 3 biscuits each. There are 100 sweets in total. We need to find how many biscuits she packed. 2. **Defi

1. **State the problem:** Polly’s mother packed party bags with 4 sweets and 3 biscuits each. There are 100 sweets in total. We need to find how many biscuits she packed. 2. **Defi

1. **State the problem:** We need to find the equation of a line that is perpendicular to the line $$y = \frac{1}{6}x + 3$$ and passes through the point $$(-3, 23)$$. 2. **Recall t

1. **State the problem:** Find the equation of the line perpendicular to $y = -\frac{2}{3} x$ that passes through the point $(4,-8)$. 2. **Recall the slope of the given line:** The

1. **State the problem:** Simplify the expression $$16^{-\frac{3}{2}}$$. 2. **Recall the rules:** For any positive number $a$ and rational exponent $m/n$, $$a^{\frac{m}{n}} = \sqrt

1. **State the problem:** We need to find the equation of an increasing S-shaped curve (sigmoid-like) with an x-intercept at $(-1, 0)$ and passing through the point $(2, 5)$. 2. **

1. **State the problem:** We are given a quartic function of the form $$y = a(x - b)^2(x - c)^2$$ and a graph with x-intercepts at $(-2,0)$ and $(1,0)$, and a point $(0,2)$ on the

1. **State the problem:** Find the equations of lines passing through the point $(10,-2)$ parallel to the given lines $4x-2y+1=0$ and $x+5y-10=0$. 2. **Find the slope of each given