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Chapter 2: Problem 19

Evaluate each expression for \(m\), applying the slope formula. \(m=\frac{6-2}{5-3}\)

Short Answer

Expert verified
The slope \(m\) is 2.

Step by step solution

01

Identify the Slope Formula

The slope formula is given by \(m = \frac{y_2-y_1}{x_2-x_1}\).
02

Identify Points in the Formula

From the given expression, identify the coordinates of the points. Here, you have \((x_1, y_1) = (3, 2)\) and \((x_2, y_2) = (5, 6)\).
03

Substitute Values into the Formula

Substitute the identified coordinates into the slope formula as follows: \(m = \frac{6-2}{5-3}\).
04

Simplify the Expression

Perform the subtraction in the numerator and the denominator: \(m = \frac{4}{2}\).
05

Evaluate the Final Answer

Divide the values: \(m = 2\).

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Key Concepts

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

Understanding Coordinates
In the context of the slope formula, coordinates are used to represent points on a Cartesian plane. Every point in this plane is given by a pair of numbers \(x_1, y_1\) and \(x_2, y_2\).

For example, when we have the points (3, 2) and (5, 6), the first number in each pair represents the x-coordinate, and the second number represents the y-coordinate.

Understanding how to organize and identify these coordinates is crucial to applying the slope formula correctly.
Numerator and Denominator Clarified
The slope formula is structured as a fraction, which consists of a numerator and a denominator.

The numerator is the part of the fraction above the line, while the denominator is below the line. In the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), the numerator is \(y_2 - y_1\) and the denominator is \(x_2 - x_1\).

This structure helps in organizing how we subtract the y-coordinates and x-coordinates from each other to find the slope.
Substitution in Equations
Substitution is the process of replacing variables with their corresponding values. In the slope formula, we substitute the coordinates of the points into the formula.

For instance, from the example given, you substitute \(y_2 = 6\), \(y_1 = 2\), \(x_2 = 5\), and \(x_1 = 3\) into \(m = \frac{y_2 - y_1}{x_2 - x_1}\):

Substituting these values, you get \ m = \frac{6-2}{5-3} \. This process allows the formula to be evaluated with actual numbers, leading to a final answer.

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Most popular questions from this chapter

Three points that lie on the same straight line are said to be collinear. Consider the points \(A(3,1), B(6,2),\) and \(C(9,3) .\) Find the slope of segment \(A C\).

Find the slope of each line, and sketch its graph. \(x+2=0\)

In \(2010,\) the number of digital cameras shipped worldwide totaled 122 million. There were 24 million shipped in \(2016 .\) Find and interpret the average rate of change in the number of digital cameras shipped worldwide per year to the nearest million.

Determine whether each pair of lines is parallel, perpendicular, or neither. The line passing through (4,6) and (-8,7) and the line passing through (-5,5) and (7,4)

An equation that defines \(y\) as a function \(f\) of \(x\) is given. (a) Solve for \(y\) in terms of \(x\), and write each equation using function notation \(f(x) .\) (b) Find \(f(3)\). $$ y-3 x^{2}=2 $$
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