🧮 algebra

1. **State the problem:** Given that $a - b = b - c = 2$, find the value of $$\frac{(a - b)^2 + (b - c)^2}{(a - c)^2}$$. 2. **Use the given information:** We know that

1. **Problem statement:** Given two functions defined by points: $f = \{(3,4), (4,5), (5,6), (6,7)\}$ and $g = \{(5,3), (6,4), (7,-2), (8,0)\}$, find:

1. The problem asks to describe the transformation applied to the graph of the function $f(x)$ when it is multiplied by 6, i.e., the function becomes $6 \cdot f(x)$. 2. The general

1. **State the problem:** We have a car purchased for 20700 dollars, and its value depreciates by 11% per year. We want to find the value of the car after 11 years.

1. **Problem:** Factor the cubic expression $x^3 + 8$ completely. 2. **Formula and rules:** This is a sum of cubes, which factors as:

1. Problem a: Find parametric equations for the solution set of $7x - 5y = 3$. 2. This is a linear equation in two variables. We can express one variable in terms of the other.

1. The problem asks us to plot the numbers $-2 \frac{1}{6}$ and $1 \frac{2}{3}$ on a number line ranging from $-3$ to $3$. 2. First, convert the mixed numbers to improper fractions

1. The problem is to solve the equation $2x + 3 = 11$ for $x$. 2. We use the basic algebraic principle of isolating the variable $x$ by performing inverse operations.

1. **State the problem:** Solve the equation $6x6 = 9x$ for $x$. 2. **Write the equation:**

1. Stating the problem: Simplify the expression $-7s(2r)(-4r^8s^2)$ and compare it to $56r^9s^3$. 2. Use the distributive property and multiplication rules for exponents: When mult

1. **State the problem:** We need to find two complementary angles where one angle is forty-four times the other. 2. **Recall the definition of complementary angles:** Two angles a

1. **State the problem:** Multiply the binomials $4w + 2$ and $7w - 6$ using the area model or distributive property. 2. **Recall the distributive property (FOIL method):**

1. **State the problem:** We are given a rectangle divided into four smaller rectangles with terms along the sides: $7z$ and $-4$ on one side, and $7z$ and $4$ on the other side. W

1. **State the problem:** Find the product of the binomials $$(7z + 4)(7z - 4)$$. 2. **Formula used:** This is a difference of squares pattern, where $$(a + b)(a - b) = a^2 - b^2$$

1. **State the problem:** We need to solve the expression $\frac{1}{5} + \left(-\frac{4}{5}\right)$. 2. **Recall the rule for adding fractions with the same denominator:** When add

1. **State the problem:** We are given the function $y = 3(x + 5)(x - 2)$ and want to understand its properties. 2. **Formula and rules:** This is a quadratic function in factored

1. **State the problem:** We are given a triangle with sides labeled $a^2 - 10$, $a^2 - 5a$, and $6$. We want to find the value(s) of $a$ that satisfy the triangle's conditions. 2.

1. Problem b: Simplify $5^{\frac{1}{4}} \cdot 5^{\frac{11}{4}}$. 2. Use the rule of exponents: $a^m \cdot a^n = a^{m+n}$.

1. **State the problem:** Simplify the expression $\left(4\sqrt{5}-3\right)\left(\sqrt{5}-2\right)$.\n\n2. **Use the distributive property (FOIL method):** Multiply each term in th

1. **State the problem:** A gopher runs from the southeast corner of a rectangular yard to the opposite corner by running the width and then the length, each 6.1 meters, and then r

1. The problem asks for the expression representing the perimeter of a rectangle with width $(10f + 8)$ cm and length $(5f - 3)$ cm. 2. Recall the formula for the perimeter $P$ of