Problem 89 - Algebra A Combined Function

Question

Find the slope of the line through the given points. $$ (14.3,-10.1) \text { and }(9.8,-2.9) $$

Step-by-Step Solution

Answer

The slope of the line is -1.6.

Step 1: Identify the Given Points

First, identify the two points given as \((x_1, y_1) = (14.3, -10.1)\) and \((x_2, y_2) = (9.8, -2.9)\).

Step 2: Recall the Slope Formula

The formula to find the slope \(m\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Step 3: Substitute the Values

Substitute the given coordinates into the slope formula. This gives:\[ m = \frac{-2.9 - (-10.1)}{9.8 - 14.3} \].

Step 4: Simplify the Numerator

Calculate the change in y-coordinates: \( -2.9 - (-10.1) = -2.9 + 10.1 = 7.2 \).

Step 5: Simplify the Denominator

Calculate the change in x-coordinates: \( 9.8 - 14.3 = -4.5 \).

Step 6: Calculate the Slope

Substitute the simplified values into the slope formula: \[ m = \frac{7.2}{-4.5} \].Simplify to find:\[ m = -1.6 \].

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that speaks the language of geometric shapes using algebra. It connects algebraic equations to geometrical figures, making it possible to quantify the geometry of shapes on the coordinate plane.

The basics of coordinate geometry involve the coordinate plane, which consists of two number lines, called axes, that intersect at right angles. These axes divide the plane into four quadrants. Each point in the plane can be defined by an ordered pair of numbers, known as coordinates. For example, the points

• (14.3, -10.1)
• (9.8, -2.9)

clearly show how we pinpoint locations in the plane. These pairs of coordinates are crucial when finding relationships such as the slope of a line between two points.

Slope Formula

The slope formula is an essential part of measuring how steep a line is on a graph. It describes the rate of change, or how much the "y" value changes with each step of the "x" value. The formula itself is quite simple:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

where:

• \( m \) represents the slope
• \( (x_1, y_1) \) and \( (x_2, y_2) \) are any two distinct points on a line

The slope tells you if the line is rising, falling, or staying constant.

• Positive slope: the line goes upwards as it moves to the right.
• Negative slope: the line goes downwards.
• Zero slope: the line is horizontal.

In the example exercise, by plugging the two sets of points into our slope formula, we find our slope to be \( m = -1.6 \), indicating a downward slope.

Linear Equation

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. In geometry, it represents a straight line on the coordinate plane.

A common form of a linear equation is the slope-intercept form, given by:

• \( y = mx + c \)

where:

• \( m \) is the slope of the line
• \( c \) is the y-intercept, the point where the line crosses the y-axis

Once we have the slope from given points, like \( m = -1.6 \) in our example, together with any chosen point, we can use this form to draw a full equation for the line. Linear equations provide a direct relationship, making them very powerful for predicting values.

Mathematics Education

Mathematics education plays a crucial role in fostering understanding and application of mathematical concepts like coordinate geometry and linear equations. It is about more than just solving for the right answer; it's about building a solid foundation of logical thinking and problem-solving skills.

Through concepts like the slope formula, students learn about change and rate, translating real-world phenomena into mathematical models. A comprehensive understanding equips students to apply these concepts to various fields, from physics to economics, and even coding.

• Enhancing problem-solving skills by tackling diverse problems and scenarios.
• Building logical reasoning.
• Promoting critical thinking.

Teachers and educators aim to provide clear explanations and relatable examples, making abstract concepts concrete and understandable for learners.