🧮 algebra

1. **Stating the problem:** Understand the concept of Ratio and solve typical ratio problems seen in competitive exams like GMAT or bank job exams. 2. **Concept of Ratio:** A ratio

1. **State the problem:** Solve the system of equations using substitution: $$y = 4x - 9$$

1. **State the problem:** Solve the system of equations using the substitution method: $$-12x - 2y = -6$$

1. **State the problem:** Write the expression $$\frac{3 + i}{2 - i} + \frac{4 + 10i}{-9 + 7i}$$ in the form $$a + ib$$ where $$a$$ and $$b$$ are real numbers. 2. **Recall the form

1. **Problem statement:** We have a relation $f = \{(5nm, 3m + 3n) : m,n \in \mathbb{Z}\}$. We want to find two pairs $(m_1, n_1)$ and $(m_2, n_2)$ such that they have the same dom

1. **Problem:** Solve the rational equation $$\frac{1}{x - 1} + 5 = \frac{11}{x - 1}$$ and find restrictions on $x$. 2. **Restrictions:** The denominator $x - 1$ cannot be zero bec

1. Problem: Solve a quadratic equation of the form $ax^2 + bx + c = 0$. 2. Formula: The solutions are given by the quadratic formula:

1. **State the problem:** Solve the quadratic equation $$x^2 - 5x - 14 = 0$$. 2. **Formula used:** The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where the equ

1. **State the problem:** We need to find the number of 1-peso coins and 25-centavo coins that total 20 coins and amount to 9.50 pesos.

1. The problem asks to create a graph similar to the described stepwise curve but different in values. 2. The original graph has three segments: an orange curve rising steeply from

1. Stating the problem: Solve the equation $$\frac{2}{3} + \frac{1}{a - b} = \frac{2}{3}a$$ for $a$. 2. Write down the equation:

1. **State the problem:** Simplify the expression $$\frac{a}{2x+2y} = \frac{3x+3y - b}{3x+3y}$$ and understand the relationship between the terms. 2. **Rewrite the denominators:**

1. The problem is to find the equation of the ellipse given its center and axes. 2. The ellipse is centered at $(3,2)$.

1. The user asked for an equation but did not specify the type or context. 2. A simple example of an equation is a linear equation, which can be written as $$y = mx + b$$ where $m$

1. **State the problem:** Mindy and Troy together ate 9 pieces of cake. Mindy ate 3 pieces, and Troy ate \(\frac{1}{4}\) of the total cake. We need to find the total number of piec

1. The problem is to add $x$ to $x$. 2. The formula for addition of like terms is $a + a = 2a$.

1. The problem asks to find the equation representing the cost $c$ of using a carpooling service for $d$ days based on the graph. 2. From the graph, the cost starts at $0$ when $d=

1. The problem states: For every $\frac{1}{4}$ mile Pascal walks, Lucien runs $\frac{2}{3}$ mile. We want to find how many miles Pascal walks for every 1 mile Lucien runs. 2. To fi

1. **Problem a:** Solve the equation $$\frac{6}{x - 1} = \frac{x}{x^2 + 2x + 1}$$ 2. **Step 1:** Recognize that $$x^2 + 2x + 1 = (x + 1)^2$$.

1. The problem is to find the decimal value of the fraction $\frac{11}{20}$. 2. The formula to convert a fraction to a decimal is to divide the numerator by the denominator: $\text

1. The problem is to find the value of the fraction $\frac{1}{4}$. 2. A fraction $\frac{a}{b}$ represents the division of $a$ by $b$.