🧮 algebra

1. **State the problem:** We are given the equation $$y - 3.5 = -2(x + 1)$$ and asked to analyze or work with it. 2. **Rewrite the equation in slope-intercept form:** The slope-int

1. 問題は数列の和 $S = \sum_{k=1}^n \frac{1}{(2k-1)(2k+1)}$ を求めることです。 2. 第$k$項は分母が積の形 $ (2k-1)(2k+1) $ になっているので、部分分数分解を使います。

1. **State the problem:** Find the equation of the line passing through the point $(5, 8)$ with slope $m = -\frac{5}{2}$. 2. **Formula:** The point-slope form of a line is given by

1. **State the problem:** Solve for $c$ in the equation $$c + 3 + 8 = 12$$. 2. **Combine like terms:** Add the constants on the left side:

1. **State the problem:** Solve the equation $$6 + 8(x - 1) = 2(3x + 4)$$ for $$x$$. 2. **Use the distributive property:** Expand both sides.

1. **State the problem:** Solve the equation $$22 + 54x = 10(6x - 2)$$ for $$x$$. 2. **Apply the distributive property:** Multiply $$10$$ by each term inside the parentheses:

1. **State the problem:** Solve the equation $$22 + 54x = (6x - 2)$$ for $$x$$. 2. **Rewrite the equation:** Remove parentheses (no change here since it's just one term inside):

1. **State the problem:** Simplify the expression $$22 + 54x + (6x - 2)$$. 2. **Apply the distributive property:** Remove the parentheses by distributing the plus sign:

1. **State the problem:** Solve the equation $$2x - 1 = \frac{3}{4}x + 9$$ for $x$. 2. **Write down the formula and rules:** To solve for $x$, we want to isolate $x$ on one side of

1. **State the problem:** Solve the linear equation $$-5x - 16 = 8x - 3$$ for $x$. 2. **Write down the formula and rules:** To solve for $x$, we want to isolate $x$ on one side of

1. **State the problem:** Solve the linear equation $$22 + 54x = -20 + 60x$$. 2. **Write down the formula and rules:** To solve for $$x$$, we want to isolate $$x$$ on one side of t

1. **State the problem:** We want to solve simultaneous equations using the elimination method. 2. **Explain the elimination method:** The elimination method involves adding or sub

1. **State the problem:** We need to sketch the graph of the function $$f(x) = 2 + 3x - x^3$$. 2. **Understand the function:** This is a cubic polynomial function of the form $$f(x

1. **State the problem:** Solve the equation $$\frac{2}{x+3} + \frac{2}{x-4} = \frac{2}{x^2 - x - 12}$$ for $x$. 2. **Factor the denominator on the right side:** Note that $$x^2 -

1. **State the problem:** We need to graph the system of linear inequalities: $$\begin{cases} y \leq -5x + 2 \\ y \leq -\frac{3}{2}x - 1 \end{cases}$$

1. **State the problem:** We are given the polynomial function:

1. **State the problem:** We need to find two supplementary angles whose difference is 42 degrees. 2. **Recall the definition:** Supplementary angles add up to 180 degrees. So if t

1. **State the problem:** We want to graph the function $$y=\frac{1}{2}x-1$$ with the restriction that $$x \ne 4$$. 2. **Understand the function:** This is a linear function with s

1. **State the problem:** Solve the equation $$\frac{6x + 7}{7} - 3 = 5x - \frac{3}{8}$$ for $$x$$. 2. **Rewrite the equation:**

1. **State the problem:** Solve the equation $$\frac{9x + 7}{2} - x + \frac{x - 2}{7} = 36$$. 2. **Identify the formula and rules:** To solve this linear equation with fractions, w

1. **State the problem:** We are given two functions $f(x) = -3x + 4$ and $g(x) = f(x) + 1$. We need to describe the transformation from the graph of $f$ to the graph of $g$. 2. **