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Chapter 3: Problem 103

Show that the slope of the line given by \(y=m x+b\) is \(m .\) (Hint: Substitute both 0 and 1 for \(x\) to find two pairs of coordinates. Then use the formula, Slope \(=\) change in \(y /\) change in \(x .\) )

Short Answer

Expert verified
The slope of the line \(y = mx + b\) is \(m\).

Step by step solution

01

- Identify the Equation of the Line

The given equation of the line is in the form of \(y = mx + b\). Here, \(m\) represents the slope, and \(b\) represents the y-intercept.
02

- Determine Coordinates for x = 0 and x = 1

Substitute \(x = 0\) into the equation to find the corresponding y-coordinate: \(y = m(0) + b = b\). So, one point is \((0, b)\). Now, substitute \(x = 1\) into the equation: \(y = m(1) + b = m + b\). The second point is \((1, m + b)\).
03

- Calculate the Change in y and the Change in x

To find the slope using two points, use the formula \(\text{Slope} = \frac{\Delta y}{\Delta x}\). Let's calculate the change in y: \(\Delta y = (m + b) - b = m\). The change in x is: \(\Delta x = 1 - 0 = 1\).
04

- Compute the Slope

The slope is found by dividing the change in y by the change in x: \(\text{Slope} = \frac{\Delta y}{\Delta x} = \frac{m}{1} = m\). Thus, we have shown that the slope of the line is \(m\).

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Key Concepts

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

Linear Equations
A linear equation describes a straight line on a graph. It is usually written as \(y = mx + b\), where:
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