Problem 31 - Understanding Elementary Algebra with Geometry

Question

Find the slope of the line passing through the given points. Round to the nearest hundredth where necessary. \((9.62,8.77)\) and \((-1.4,8.77)\)

Step-by-Step Solution

Answer

The slope is 0.

Step 1: Recall the slope formula

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

Step 2: Substitute the given points into the formula

Given points are \( (9.62, 8.77) \) and \( (-1.4, 8.77) \). Substituting these points into the formula: \[ m = \frac{8.77 - 8.77}{-1.4 - 9.62} \]

Step 3: Simplify the numerator

Calculate the numerator: \[ 8.77 - 8.77 = 0 \]

Step 4: Simplify the denominator

Calculate the denominator: \[ -1.4 - 9.62 = -11.02 \]

Step 5: Compute the slope

Perform the division: \[ m = \frac{0}{-11.02} = 0 \]

Key Concepts

Slope Formula

The slope of a line indicates its steepness and is essential in understanding linear equations and algebraic functions. The slope formula is given by: \( m = \frac{y_2 - y_1}{x_2 - x_1} \) where \( (x_1, y_1) \) and \( (x_2, y_2) \) are coordinates of any two points on the line. This formula calculates the ratio of the vertical change (difference in y-coordinates) to the horizontal change (difference in x-coordinates) between two points. \
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This calculation helps determine whether a line is rising, falling, or remains constant as it moves from left to right \

Coordinates

Coordinates are pairs of numbers that define a point's location on a plane, typically represented as (x, y). The first number, x, denotes horizontal position while the second number, y, specifies the vertical position. \
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In the given problem, the points \((9.62, 8.77)\) and \((-1.4, 8.77)\) are used. Both of these points share the same y-coordinate, 8.77, indicating that the line passes through these points horizontally because there is no vertical change. \
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Understanding these points in the coordinate system is fundamental to determining how a line behaves.

Linear Equations

Linear equations form the foundation of many algebraic concepts. An equation is called linear if it forms a straight line when plotted on a graph. The standard format for linear equations is \(y = mx + c\), where: \

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• y represents the dependent variable \
• x is the independent variable \
• m is the slope \
• c is the y-intercept, the point where the line crosses the y-axis\

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Identifying the slope from two points helps in building and understanding linear equations. For our particular problem, the fact that the slope \((m)\) is zero indicates a horizontal line, emphasizing no vertical change as x varies.

Algebraic Functions

Algebraic functions involve operations on variables, and in this context, we focus on linear functions, a subset of algebraic functions. These functions are crucial as they describe relationships with constant rates of change. \
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The slope formula, \( m = \frac{y_2 - y_1}{x_2 - x_1} \), embodies these concepts by measuring this constant rate, or the gradient of the line. \
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In our problem, solving the slope provides insight into the relationship defined by the linear function passing through \((9.62, 8.77)\) and \((-1.4, 8.77)\). This particular relationship is constant since there is no vertical shift across these points, asserting the zero slope.