🧮 algebra

Let's explore the formula step by step! 🎉 1. Imagine you have some numbers called $D_n$ and $D_{n-1}$.

1. **State the problem:** Solve for $x$ in the equation $$\log_{2^x} 16 + 3 \log_{3^x} 8 = 1.$$\n\n2. **Recall the logarithm change of base formula:** For any positive $a,b,c$ with

1. The user requests to graph the given functions and the circle equation. 2. The functions are:

1. The problem is to find the sum of integers from 1 to $n$ using the given flowchart algorithm. 2. The algorithm initializes $\text{sum} = 0$ and $a = 1$.

1. **State the problem:** Ingrid works 35 hours per week at a rate of 11.82 per hour. She is paid overtime at time and a half for any hours worked beyond 35 hours. Last week, she w

1. The problem asks for the gradients (slopes) of two lines, J and K, given their equations. 2. The general form of a straight line is $y = mx + c$, where $m$ is the gradient (slop

1. **Problem Statement:** We are given data for Fruit A and Fruit B showing the number of fruits purchased, total weight, and total price. We need to determine if the price of Frui

1. **State the problem:** Simplify or analyze the expression $d^2 + 8d^3r^5 + 15rx^2$. 2. **Identify terms:** The expression consists of three terms: $d^2$, $8d^3r^5$, and $15rx^2$

1. **State the problem:** Factor the expression $$Dx^6 + Dx^3 r x^5 + 15 r x^{10}$$. 2. **Rewrite the expression:** Combine like terms and powers of $x$:

1. **State the problem:** Factor the quadratic expression $2x^2 - 6x - 8$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multip

1. **State the problem:** Solve the system of linear equations: $$4x - 5y = -3$$

1. **State the problem:** Solve the system of linear equations: $$4x + 5y = 6$$

1. **State the problem:** Solve the system of equations: $$\frac{1}{2}x + \frac{1}{3}y = 4$$

Let's find the value of $y$ when $x = 2$ in the equation $y = 2x + 3$. **Step 1:** Imagine you have 2 groups of toys. Each group has 2 toys because of $2x$.

1. **State the problem:** Solve the system of equations: $$\begin{cases} x + y = 3 \\ x - y = -1 \end{cases}$$

1. The problem involves solving the system of linear equations: $$\begin{cases} 7x - 8y = 45 \\ 8x - 7y = 88 \end{cases}$$

Let's add two 2 by 2 matrices step by step! 🎉 Imagine you have two groups of toys arranged in squares. Each square has some toys. We will add the toys in the same spots!

1. **State the problem:** Find the roots of the equation $$x^3 - 4x^2 + x + 14 = 8$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero:

1. **State the problem:** Find the roots of the polynomial equation $$4x^3 + 7x^2 - 5x - 6 = 0$$ using synthetic division. 2. **Recall the Rational Root Theorem:** Possible rationa

1. **State the problem:** Solve the equation $$3(7z - 23) = 99$$ for the variable $z$. 2. **Use the distributive property:** Multiply 3 by each term inside the parentheses:

1. Problem: A rectangular rug has an area of 60 in.² and a perimeter of 34 in. We need to find its dimensions. 2. Let the length be $l$ and the width be $w$. The formulas are: