🧮 algebra

1. **State the problem:** We are given the ratio of 12-year-olds to 13-year-olds as $5:3$ and the total number of students as $24$. We need to find how many students are 13 years o

1. **State the problem:** We need to find the equation of a line that passes through the point $(6, -4)$ and is perpendicular to the line given by the equation $$6 = 4y + 3x$$. 2.

1. **State the problem:** Find the equation of the line passing through points $(3,-5)$ and $(-2,7)$. 2. **Formula used:** The point-slope form of a line is:

1. **State the problem:** Simplify the expression $$\frac{2x-3x}{4y} \cdot \frac{y}{x^2-9}$$. 2. **Combine and simplify the numerator in the first fraction:**

1. **State the problem:** Solve the quadratic equation $$2x^2 - 4x + 5 = 0$$. 2. **Formula used:** The quadratic formula for solving $$ax^2 + bx + c = 0$$ is $$x = \frac{-b \pm \sq

1. **State the problem:** Solve the equation $$\frac{3x}{2x-1} - \frac{2}{5-x} = 7.$$\n\n2. **Find a common denominator:** The denominators are $2x-1$ and $5-x$. The common denomin

1. **State the problem:** Solve for $x$ in the equation $0.06x = 1691$. 2. **Formula and rules:** To isolate $x$, divide both sides of the equation by $0.06$.

1. **State the problem:** Solve the equation $$8 = 45y$$ for the variable $$y$$. 2. **Formula and rules:** To isolate $$y$$, divide both sides of the equation by $$45$$.

1. **State the problem:** Solve the equation $$\frac{2}{3}r = 4$$ for the variable $r$. 2. **Formula and rules:** To isolate $r$, multiply both sides of the equation by the recipro

1. **State the problem:** Solve the equation $$12r=30$$ for the variable $$r$$. 2. **Formula and rules:** To isolate $$r$$, divide both sides of the equation by $$12$$. Remember, d

1. **State the problem:** Solve the equation $$-72 = 18v$$ for the variable $$v$$. 2. **Formula and rules:** To isolate $$v$$, divide both sides of the equation by the coefficient

1. **State the problem:** Solve the equation $$-3y=0$$ for the variable $y$. 2. **Formula and rules:** To solve for $y$, we need to isolate $y$ on one side of the equation. Since $

1. **State the problem:** Solve the equation $-2r=16$ for $r$. 2. **Formula and rules:** To solve for $r$, divide both sides of the equation by the coefficient of $r$, which is $-2

1. **State the problem:** Solve the equation $$3r = 9$$ for $$r$$. 2. **Formula and rules:** To solve for $$r$$, divide both sides of the equation by the coefficient of $$r$$, whic

1. **State the problem:** We have two exponential functions $f$ and $g$ defined on the interval $[0,4]$. Function $f$ is given by a table of values, and function $g$ passes through

1. **State the problem:** We are given the quadratic equation $x^2 + 10x + 27 = 0$ and need to complete the square to rewrite it in the form $(x + a)^2 = b$. 2. **Recall the formul

1. **State the problem:** Simplify the expression $$\frac{a^3b^5}{a^4b}$$ to find an equivalent expression. 2. **Recall the laws of exponents:**

1. **State the problem:** We need to determine the end behavior of the function $$f(x) = 3|x - 7| - 7$$ as $x$ approaches positive and negative infinity. 2. **Recall the properties

1. **State the problem:** We are given the expression $$3^{8/5}$$ and asked to rewrite it in the form $$\sqrt[5]{a^b}$$, where $$a$$ and $$b$$ are numbers to find. 2. **Recall the

1. **State the problem:** We have a function $f$ and want to find the function $g$ that results from translating $f$ down by 4 units. 2. **Recall the translation rule:** Translatin

1. **State the problem:** Factor the quadratic expression $$9x^2 + 3x - 2$$. 2. **Recall the factoring method:** For a quadratic $$ax^2 + bx + c$$, we look for two binomials $$(mx