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Chapter 3: Problem 85

In the following exercises, use the slope formula to find the slope of the line between each pair of points. $$ (2,5),(4,0) $$

Short Answer

Expert verified
The slope is \(m = -\frac{5}{2}\).

Step by step solution

01

Understand the Slope Formula

The slope (m) between two points \(x_1, y_1\) and \(x_2, y_2\) is given by the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].
02

Identify the Points

For the points (2, 5) and (4, 0), assign \(x_1 = 2\), \(y_1 = 5\), \(x_2 = 4\), and \(y_2 = 0\).
03

Plug in the Values

Substitute the values into the formula: \[ m = \frac{0 - 5}{4 - 2} \].
04

Simplify the Expression

Execute the subtraction in the numerator and the denominator: \[ m = \frac{-5}{2} \].
05

Interpret the Slope

The slope of the line that goes through points (2, 5) and (4, 0) is \(m = -\frac{5}{2}\). This means that for every 2 units you move to the right on the x-axis, the value of y decreases by 5 units.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a method used to study geometry using a coordinate system. Here, we connect algebra to geometry through graphs and equations. By plotting points on the xy-plane and understanding their relationships, it becomes easier to analyze the properties and behavior of geometric shapes. It's a powerful tool because:
  • It helps us visualize algebraic equations by graphing them.
  • We can determine distances and midpoints between points.
  • It allows us to find slopes and equations of lines.
For example, plotting the points (2, 5) and (4, 0) on a graph helps us see the line between them, and using coordinate geometry we can easily calculate that line's slope.
Finding Slope
Finding the slope of a line is a fundamental skill in algebra and coordinate geometry. The slope tells us how steep a line is and the direction it goes.

The slope is calculated by the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula shows the change in y-coordinates divided by the change in x-coordinates between two points.
  • If the slope is positive, the line goes upwards as you move from left to right.
  • If the slope is negative, the line goes downwards.
  • A zero slope means the line is horizontal.
  • An undefined slope means the line is vertical.
For the given points (2, 5) and (4, 0), substituting into the formula gives us \( m = \frac{0 - 5}{4 - 2} = \frac{-5}{2} \). This means the line decreases by 5 units in y for every 2 units it increases in x.
Linear Equations
In algebra, a linear equation is any equation that can be written in the form \( y = mx + b \) where m is the slope and b is the y-intercept. This form is very useful because:
  • It lets you easily graph linear equations.
  • Knowing m and b, you can quickly sketch the line.
  • It's straightforward to understand the relationship between x and y.
For our slope \( m = -\frac{5}{2} \), a possible linear equation of the line passing through point (2, 5) would be found by substituting the point and slope into the point-slope form \( y - y_1 = m(x - x_1) \):
\[ y - 5 = -\frac{5}{2}(x - 2) \]. When simplified, this equation describes the line passing through (2, 5) and (4, 0).
Algebra II
Algebra II takes the concepts from Algebra I and expands them to more complex scenarios. It covers a wide array of topics including:
  • Quadratic equations
  • Polynomials
  • Radicals
  • Exponential and logarithmic functions
The slope formula is fundamental in Algebra II as it prepares you for more advanced mathematics. Understanding how to find slopes and interpret them ties into graphing different types of functions and learning about their properties. Mastering these concepts is crucial for tackling more advanced problems, and it strengthens your analytical and problem-solving skills. Always practice finding the slope and working with linear equations to build a solid foundation for Algebra II.

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