🧮 algebra

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

1. **State the problem:** We have a path represented by the equation $$y = -2x + 5$$ and a new path perpendicular to it passing through the point $$(-2, 9)$$. We need to find the e

1. **State the problem:** We have the equation $$y = 40x + 55$$ where $y$ is the total amount of money spent and $x$ is the number of chairs bought. 2. **Identify the y-intercept:*

1. **State the problem:** We need to determine which ordered pair $(x, y)$ satisfies the inequality $$y > 2x - 3$$. 2. **Recall the inequality:** The inequality means the $y$-value

1. **State the problem:** We need to find the point-slope form of a line passing through the point $(-2, 4)$ with slope $3$. 2. **Recall the point-slope formula:** The point-slope

1. **State the problem:** Convert the equation $y + 5 = 4(x + 8)$ into slope-intercept form, which is $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept. 2. **Use the d

1. **State the problem:** We need to find the equation of a line parallel to $y = 5x - 4$ that passes through the point $(3, 4)$. 2. **Recall the rule for parallel lines:** Lines a

1. **State the problem:** We need to find the composition $f \circ g \circ h$, which means $f(g(h(x)))$. 2. **Recall the functions:**

1. **Problem 10:** Write the standard form of the circle. The standard form of a circle is given by:

1. **State the problem:** We need to find the values of $x$ such that $f(x) = 1$ for the given quadratic function. 2. **Understand the graph:** The parabola opens upward with verte

1. **Problem Statement:** Prove that the set $\{e^{ax}, e^{bx}\}$ is linearly independent on $\mathbb{R}$. 2. **Definition:** A set of functions $\{f_1, f_2\}$ is linearly independ

1. **State the problem:** We need to find the value(s) of $x$ such that $f(x) = -7$ for the given linear function. 2. **Analyze the graph:** The graph shows a straight line with a

1. **State the problem:** Solve the quadratic equation $$9z^2 + 16 = 0$$. 2. **Rewrite in standard form:** The standard form is $$az^2 + bz + c = 0$$. Here, $$a=9$$, $$b=0$$, and $

1. **State the problem:** We need to sketch the graph of the piecewise function:

1. **State the problem:** Taylor wants to find 5 times any number by first multiplying the number by 10 and then taking half of the result. 2. **Formula used:** To find 5 times a n

1. **State the problem:** Jordan needs to mix whole wheat flour and bread flour in a ratio of $1 : 2$ to get a total of $4 \frac{1}{2}$ cups of flour. 2. **Convert mixed number to

1. **State the problem:** Simplify the expression $$ (p - 2) \cdot (p - 2) \cdot (p - 2) $$. 2. **Formula used:** This is the cube of a binomial, which can be written as $$ (a - b)

1. **State the problem:** Calculate the product of the numbers $8 \cdot (-2) \cdot (-2) \cdot 8 \cdot (-2) \cdot 8 \cdot 8$. 2. **Recall the multiplication rules:**

1. The problem asks us to sketch the graph of the function $y = x^3$. 2. The function $y = x^3$ is a cubic function, which means it is a polynomial of degree 3.

1. **Problem Statement:** Find the slope and y-intercept for lines 37-40, then find intercepts for lines 41-44. ---

1. **State the problem:** We need to find the product of $-\frac{1}{3}$ and $2$. 2. **Formula used:** Multiplication of a fraction by a whole number is done by multiplying the nume