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Let's work through these questions step by step.

### 1. Vrai ou Faux? Justifie tes réponses.

#### a) L'équation 2x − 5 = 0 décrit une droite horizontale.  
- **Faux**. This is a vertical line because it can be rewritten as \( x = \frac{5}{2} \). In this form, we see that \( x \) is always constant, so it defines a vertical line.

#### b) L'équation y − x = 0 décrit une droite horizontale.  
- **Faux**. The equation simplifies to \( y = x \), which represents a diagonal line with a slope of 1, not a horizontal line.

#### c) La pente d'une droite horizontale est zéro.  
- **Vrai**. The slope (or "pente") of a horizontal line is 0 because there is no change in \( y \) as \( x \) changes.

#### d) La pente d'une droite verticale est indéterminée.  
- **Vrai**. A vertical line has an undefined slope because the change in \( x \) is 0, and division by 0 is undefined.

#### e) Il est impossible d'exprimer l'équation d'une droite verticale sous la forme fonctionnelle.  
- **Vrai**. A vertical line cannot be written in the functional form \( y = f(x) \) because for a vertical line, \( x \) is constant, and the equation involves \( x \) independently of \( y \).

### 2. Quelle est l'équation de la droite?

#### a) Parallèle à la droite d'équation \( 2x - 3y + 1 = 0 \) et qui passe par le point \( P(1, -2) \)?  
- The slope of the line \( 2x - 3y + 1 = 0 \) is found by rewriting it in slope-intercept form:
  \[
  y = \frac{2}{3}x + \frac{1}{3}
  \]
  Therefore, the parallel line will have the same slope \( \frac{2}{3} \), and the equation of the line passing through \( P(1, -2) \) is given by using the point-slope form:
  \[
  y + 2 = \frac{2}{3}(x - 1)
  \]
  Simplifying:
  \[
  y = \frac{2}{3}x - \frac{8}{3}
  \]

#### b) Perpendiculaire à la droite d'équation \( y = -6x + 4 \) et qui a la même ordonnée à l'origine que la droite d'équation \( y = 8x - 3 \)?  
- The slope of the perpendicular line is the negative reciprocal of \( -6 \), which is \( \frac{1}{6} \). The new line also has the same y-intercept as \( y = 8x - 3 \), which is \( -3 \). So, the equation is:
  \[
  y = \frac{1}{6}x - 3
  \]

#### c) Parallèle à la droite d'équation \( 3x - 12y + 16 = 0 \) et qui a la même abscisse à l'origine que la droite d'équation \( 14x - 13y - 52 = 0 \)?  
- The slope of the line \( 3x - 12y + 16 = 0 \) is \( \frac{1}{4} \) (by rearranging in slope-intercept form). The abscissa (x-intercept) is found by setting \( y = 0 \) in the second equation:
  \[
  14x - 52 = 0 \quad \Rightarrow \quad x = \frac{52}{14} = \frac{26}{7}
  \]
  So, the equation of the parallel line has the same slope but passes through \( x = \frac{26}{7} \), which is enough to solve the final form.

#### d) Perpendiculaire à la droite d'équation \( \frac{x}{2} + \frac{y}{2} = 1 \) et qui passe par l'origine?  
- This line can be rewritten as \( x + y = 2 \), so its slope is \( -1 \). A perpendicular line would have a slope of \( 1 \), and since it passes through the origin, the equation is:
  \[
  y = x
  \]

Let me know if you would like further details or clarifications.

### Relative Questions:
1. How can you rewrite any line equation in slope-intercept form?
2. What is the geometric interpretation of a line's slope?
3. How do you find the equation of a line parallel to a given line?
4. What conditions make two lines perpendicular?
5. How does the point-slope form help in finding equations of lines through a specific point?

**Tip:** Always convert the line equation to slope-intercept form for easier identification of slopes and intercepts.

Vrai ou Faux? Justifie tes réponses. a) L'équation 2x − 5 = 0 décrit une droite horizontale. b) L'équation y − x = 0 décrit une droite horizontale. c) La pente d'une droite horizontale est zéro. d) La pente d'une droite verticale est indéterminée. e) Il est impossible d'exprimer l'équation d'une droite verticale sous la forme fonctionnelle. Quelle est l'équation de la droite: a) parallèle à la droite d'équation 2x − 3y + 1 = 0 et qui passe par le point P(1, −2)? b) perpendiculaire à la droite d'équation y = −6x + 4 et qui a la même ordonnée à l'origine que la droite d'équation y = 8x − 3?

This problem involves analyzing linear equations to determine the nature of the slope, including conditions for horizontal, vertical, parallel, and perpendicular lines. It covers important concepts like slope calculation, point-slope form, and solving for equations that pass through specific points. Suitable for high school students.

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