What is the slope of the line passing through the points 2 1...

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--- ## Question 1 **Question content**: What is the slope of the line passing through the points \((-2, 1)\) and \((3, 1)\) (as shown in the coordinate grid)? Simplify the answer as a proper fraction, improper fraction, or integer. **Discipline**: Mathematics **Grade**: 8th grade (or equivalent middle school level, covering linear equations and slope) ### Correct answer \( 0 \) ### Detailed problem-solving steps 1. **Recall the slope formula**: For two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope \( m \) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] 2. **Identify the coordinates**: Let \((x_1, y_1) = (-2, 1)\) and \((x_2, y_2) = (3, 1)\). 3. **Calculate the change in \( y \) (Δ\( y \))**: \( y_2 - y_1 = 1 - 1 = 0 \) 4. **Calculate the change in \( x \) (Δ\( x \))**: \( x_2 - x_1 = 3 - (-2) = 3 + 2 = 5 \) 5. **Compute the slope**: Substitute Δ\( y \) and Δ\( x \) into the slope formula: \[ m = \frac{0}{5} = 0 \] ### Knowledge points involved 1. **Slope of a line** - Definition: The slope \( m \) of a line between two points \((x_1, y_1)\) and \((x_2, y_2)\) is the ratio of the vertical change (Δ\( y = y_2 - y_1 \)) to the horizontal change (Δ\( x = x_2 - x_1 \)), i.e., \( m = \frac{\Delta y}{\Delta x} \). - Interpretation: A slope of \( 0 \) indicates a horizontal line (constant \( y \)-value), while a undefined slope (Δ\( x = 0 \)) indicates a vertical line. 2. **Horizontal lines** - Definition: A horizontal line has the same \( y \)-coordinate for all points on the line (e.g., \( y = k \) for some constant \( k \)). - Slope property: Horizontal lines always have a slope of \( 0 \) because the vertical change (Δ\( y \)) between any two points is \( 0 \). 3. **Coordinate geometry (ordered pairs)** - Definition: An ordered pair \((x, y)\) represents a point in a coordinate plane, where \( x \) is the horizontal (x-axis) coordinate and \( y \) is the vertical (y-axis) coordinate. ---

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