🧮 algebra

1. **State the problem:** We are given a parabola with a focus at $ (3, 6) $ and a directrix $ y = -4 $. We need to find the equation of the parabola and describe its properties. 2

1. **State the problem:** Kareem mixes lemonade with a ratio of 3 cups lemon juice to 4 cups water. He uses 32 cups of water and we need to find how many cups of lemon juice are in

1. **State the problem:** Solve for $x$ in the equation $$\frac{7}{3}x - 5 = -\frac{7}{4}x - \frac{1}{2}.$$\n\n2. **Write down the equation:** $$\frac{7}{3}x - 5 = -\frac{7}{4}x -

1. **State the problem:** Solve for $y$ in the equation $$7(y - 9) - 2 = -7(-4y + 7) - 9y.$$\n\n2. **Apply the distributive property:** Multiply out the parentheses on both sides.\

1. **State the problem:** We are given the distance functions for a minivan and a pickup truck as functions of time in hours, $x$. The minivan's distance is given by $y=45x$. We ne

1. **Problem Statement:** Find the average rate of change of the sheep's mass between $t=103$ and $t=111$ days, where $M(t) = 27.6 + 0.3t - 0.001t^2$.

1. Stating the problem: Simplify the expression $$\frac{\sqrt{-6}}{\sqrt{-3} \times \sqrt{-5}}$$ and write the result in the form $a + bi$ where $a$ and $b$ are real numbers. 2. Re

1. **State the problem:** Solve the linear equation $2y + 2x = -10$ for $y$. 2. **Formula and rules:** To solve for $y$, isolate $y$ on one side of the equation by performing algeb

1. **State the problem:** We need to solve the system of equations by graphing: $$y = -\frac{6}{5}x$$

1. **State the problem:** We are given the linear equation $$Y = 52.50 - 0.85X$$ and asked to find the value of $X$ when $Y=6$ and when $Y=4$. 2. **Formula and approach:** To find

1. **State the problem:** Solve the system of linear equations: $$\begin{cases} 2x + 6y - z = 6 \\ 4x - 3y + 5z = -5 \\ 6x + 9y - 2z = 11 \end{cases}$$

1. **State the problem:** Simplify the algebraic expression $5w^2 + 3w + 2w^2 - 2w$. 2. **Group like terms:** Combine terms with $w^2$ and terms with $w$ separately.

1. **State the problem:** Simplify the expression $6(3n + 8)$. 2. **Use the distributive property:** Multiply each term inside the parentheses by 6. The distributive property state

1. **State the problem:** A boat travels 336 miles downstream and back upstream. The downstream trip takes 12 hours, and the upstream trip takes 14 hours. We need to find the speed

1. **State the problem:** Matt and Ming sold small and large boxes of oranges at a fundraiser. We know:

1. **State the problem:** We have the profit function $$P(x) = 5x^2 - 5x + 9$$ and need to find the average rate of change of profit between certain values of $x$, and the instanta

1. **State the problem:** We need to find which expressions are equivalent to $$3^{-9} \cdot 3^{-3}$$. 2. **Use the property of exponents:** When multiplying powers with the same b

1. The problem asks to evaluate the expression $$\sqrt{-9} - 6\sqrt{-2}$$ and write the result in the form $a + bi$ where $a$ and $b$ are real numbers. 2. Recall that for any negat

1. **State the problem:** We need to identify the linear inequality represented by a graph with a solid line of slope $\frac{1}{3}$ and y-intercept $-1$, with the region shaded bel

1. **State the problem:** We are given a function $f(x) = x$ and a graph of another function $g(x)$ which is a transformation of $f(x)$. We need to find the equation for $g(x)$ fro

1. **State the problem:** We have multiple linear equations of the form $h = mx + b$ with slopes $3$, $-3$, and $1$, and intercepts $1$ and $7$. We want to identify which pairs of