🧮 algebra

1. **Énoncé du problème :** On a une application linéaire $f : \mathbb{R}^3 \to \mathbb{R}^3$ définie par

1. **State the problem:** We need to find the value of $m$ such that the line $y = mx - 7$ is parallel to the line $2x + 3y = 6$. 2. **Recall the rule for parallel lines:** Two lin

1. **Problem:** Find the x and y-intercepts of the quadratic function $f(x) = x^2 + 3x - 4$. 2. **Formula and rules:**

1. The problem asks us to find which equation represents a line parallel to the line given by $$y = -3x + 4$$. 2. Recall that parallel lines have the same slope.

1. **Problem:** Graph and analyze the quadratic function $$f(x) = -x^2 - 2x + 3$$. 2. **Formula and rules:** The vertex form of a quadratic is $$f(x) = a(x-h)^2 + k$$ where $$(h,k)

1. **State the problem:** Solve the inequality $9x - 7i > 3(3x - 7u)$ for $x$. 2. **Expand the right side:**

1. **State the problem:** Solve the inequality $9x - 7we > 3(3x - 7u)$. 2. **Apply the distributive property:** Expand the right side:

1. **State the problem:** Solve the equation $$\log_9 \sqrt{x} = \frac{1}{2 \log_3 3} + \log_9 (4x^3), \quad x > 0.$$\n\n2. **Recall logarithm rules:**\n- Change of base: $$\log_a

1. The problem is to solve the exercises provided in the image. 2. Since the image content is not readable as text, please provide the specific exercises or type them out.

1. **State the problem:** Solve the exponential equation $$\left(\sqrt{3} - 1\right)^{5x+1} = \left( \frac{\sqrt{3} + 1}{2} \right)^4.$$\n\n2. **Recall important rules:** To solve

1. **State the problem:** Solve the inequality $4n^{\frac{1}{2}} + n \leq 2n$ for $n$. 2. **Rewrite the inequality:**

1. **State the problem:** We need to find the cubic function $f(x)$ such that: - $f(x)$ touches the x-axis at the origin $(0,0)$.

1. **State the problem:** Solve the inequality $1 \leq n - 4\sqrt{n}$ for $n \geq 0$ since $\sqrt{n}$ is defined only for non-negative $n$. 2. **Rewrite the inequality:**

1. **State the problem:** Solve the equation $$(\sqrt{3} - 1)^{5x + 1} = \left(\frac{\sqrt{3} + 1}{2}\right)^4.$$\n\n2. **Recall the formula and rules:** When bases are related, ex

1. **Problem:** Find the sum of the series $$\frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \dots$$ 2. **Understanding the series:** Each term is of the form $$\frac{n}{n+

1. **State the problem:** Solve the equation $x(x+2) + y(y+2) = 2000$ for possible values of $x$ and $y$. 2. **Rewrite the equation:** Expand the terms:

1. Problem: Find the values of $a$ and $b$ if $f(x) = ax^3 + bx^2 + 12$ can be divided exactly by both $(x + 1)$ and $(x - 2)$. 2. Since $(x + 1)$ and $(x - 2)$ are factors, $f(-1)

1. The problem is to simplify the expression $A=3|1-2x|-5$. 2. The absolute value function $|y|$ returns the distance of $y$ from zero, so it is always non-negative.

1. Let's start by understanding the problem: You want to know what it means to isolate $x$, perform inverse operations, and subtract from both sides in an equation. 2. **Isolating

1. The problem is to solve the equation $2x + 3 = 11$ for $x$. 2. The formula used here is to isolate $x$ by performing inverse operations.

1. **State the problem:** Solve the expression $4x + 8$ for $x$ if it is set equal to zero or another value. 2. **Formula and rules:** To solve a linear equation like $4x + 8 = 0$,