Problem 4 - Understanding Elementary Algebra with Geometry

Question

Find the slope of the line passing through the given points. Round to the nearest hundredth where necessary. \((5,1)\) and \((1,9)\)

Step-by-Step Solution

Answer

The slope is \(-2\).

Step 1 - Identify the points

The given points are \((5, 1)\) and \((1, 9)\). Label them as \((x_1, y_1)\) and \((x_2, y_2)\) respectively. So, \((x_1, y_1) = (5, 1)\) and \((x_2, y_2) = (1, 9)\).

Step 2 - Recall the slope formula

The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Step 3 - Substitute the coordinates into the formula

Substitute the coordinates of the points \((x_1, y_1) = (5, 1)\) and \((x_2, y_2) = (1, 9)\) into the slope formula: \[ m = \frac{9 - 1}{1 - 5} \]

Step 4 - Simplify the expression

Calculate the numerator and the denominator separately: \[ 9 - 1 = 8 \] \[ 1 - 5 = -4 \] Now, substitute these values back into the expression: \[ m = \frac{8}{-4} \]

Step 5 - Find the slope

Divide 8 by -4 to find the slope: \[ m = -2 \]

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry. It uses coordinates to represent points on a plane. A coordinate system typically has two axes: the x-axis (horizontal) and the y-axis (vertical). Points are represented as \( (x, y) \) where \( x \) and \( y \) are values on the x-axis and y-axis, respectively. For example, the point \( (5, 1) \) tells us that it is at 5 on the x-axis and 1 on the y-axis.

Slope Formula

The slope of a line measures its steepness. In coordinate geometry, the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. This formula measures the change in y divided by the change in x, also known as 'rise over run'. For example, to find the slope of the line passing through \( (5, 1) \) and \( (1, 9) \), we plug the values into the formula: \[ m = \frac{9 - 1}{1 - 5} = \frac{8}{-4} = -2 \].

Algebraic Calculation

Algebraic calculations are critical in finding the slope. After identifying the points and recalling the formula, substitute the coordinates: \( m = \frac{9 - 1}{1 - 5} \). The numerator and denominator must be calculated separately. Firstly, \( 9 - 1 = 8 \) results in the numerator. Secondly, \( 1 - 5 = -4 \) results in the denominator. Finally, substitute these results back: \[ m = \frac{8}{-4} \].

Mathematical Operations

Performing mathematical operations correctly is crucial for slope calculation. Operations include subtraction and division. After calculating the numerator and the denominator, you divide them: \[ 8 \div (-4) = -2 \]. This gives a slope of -2, indicating the line descends. Mathematical operations must be precise, especially when signs (positive or negative) are involved.