🧮 algebra

1. **State the problem:** Solve the simultaneous equations: $$2x^2 + 3xy + 4y^2 = 6$$

1. **Stating the problem:** We want to find how to get the numbers 4, 16, 41, 48, and 66 from the sum of 131 using numbers 1 through 10. 2. **Understanding the problem:** The probl

1. **Problem Statement:** Prove by mathematical induction that $$\sum_{r=1}^n \binom{r}{1} = \binom{n+1}{2}$$ for all positive integers $n$. 2. **Recall the definition:** $$\binom{

1. **State the problem:** Simplify or analyze the expression $1 + e^{-t} - e^{-\sin 2t}$. 2. **Recall the components:** The expression involves exponential functions with negative

1. **Énoncé du problème :** On considère la suite $(u_n)$ définie par $u_0=0$ et $u_{n+1} = \frac{u_n - 2}{2u_n + 5}$ pour tout $n \in \mathbb{N}$.

1. **Problem:** Find the equation of the axis of symmetry and the coordinates of the vertex of the parabola. 2. **Formula and rules:**

1. The problem asks for the equation of the axis of symmetry and the coordinates of the vertex of a parabola. 2. The axis of symmetry of a parabola in vertex form $y = a(x-h)^2 + k

1. **State the problem:** We want to find the time $t$ when the ball hits the ground. This means finding $t$ such that the height $h(t) = 0$. 2. **Write the height function:** The

1. **State the problem:** Simplify the expression $$\frac{x + 1}{x(x - 1)} - \frac{1}{2(x - 1)}$$ and identify which of the given options it equals. 2. **Find a common denominator:

1. **Problem:** Find the expanded form of $$\sum_{i=2}^5 2^{i-1}$$. 2. **Formula:** The summation $$\sum_{i=m}^n a_i$$ means adding terms from $$i=m$$ to $$i=n$$.

1. **State the problem:** Simplify the expression $$\frac{x}{x-3} + \frac{2}{x^2-3}$$. 2. **Identify the denominators:** The denominators are $x-3$ and $x^2-3$.

1. The problem is to understand how to solve fractions, which means simplifying or performing operations like addition, subtraction, multiplication, or division with fractions. 2.

1. **State the problem:** Simplify the expression $$3 \frac{1}{8} \left( \frac{1}{5} - 12a \right) - 1 \frac{3}{4} \left( \frac{2}{5} a + \frac{5}{7} \right)$$. 2. **Convert mixed

1. **Problem statement:** Find the distance between the two parallel lines given by the equations $$4x + 3y - 2 = 0$$ and $$4x + 3y + 8 = 0$$. 2. **Formula used:** The distance $d$

1. **State the problem:** Solve the linear equation $3x + 2 = 5$ for $x$. 2. **Formula and rules:** To solve for $x$, isolate the variable by performing inverse operations. Subtrac

1. The problem is to simplify the expression $x2*4p$. 2. First, recognize that $x2$ means $x \times 2$ and $4p$ means $4 \times p$.

1. **Stating the problem:** We want to add two fractions representing parts of a pizza, for example, $\frac{1}{4}$ and $\frac{1}{3}$.\n\n2. **Formula and rules:** To add fractions,

1. **Problem:** draw a fraction pizza image. 2. Since you asked to draw it, I will make a simple pizza-style fraction diagram.

1. Let's solve a fraction problem: Simplify the expression $$\frac{3}{4} + \frac{5}{6}$$. 2. To add fractions, we need a common denominator. The denominators here are 4 and 6.

1. **State the problem:** We need to calculate the sum and difference of the fractions $\frac{4}{5} + \frac{3}{20} - \frac{1}{10}$ and simplify the result. 2. **Find a common denom

1. **State the problem:** Calculate $\frac{6}{8} - \frac{3}{5}$ and simplify the result. 2. **Find a common denominator:** The denominators are 8 and 5. The least common denominato