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Chapter 8: Problem 66

Find the coordinates of two points on the given line, and then use those coordinates to find the slope of the line. $$y=6 x$$

Short Answer

Expert verified
Two points are (0, 0) and (1, 6); slope is 6.

Step by step solution

01

Choose Values for x

To find coordinates on the line, we start by selecting values for \( x \). Let's choose \( x = 0 \) and \( x = 1 \).
02

Calculate y for x = 0

Substitute \( x = 0 \) into the equation \( y = 6x \). This gives \( y = 6(0) = 0 \). So, one point is \((0, 0)\).
03

Calculate y for x = 1

Substitute \( x = 1 \) into the equation \( y = 6x \). This gives \( y = 6(1) = 6 \). So, another point is \((1, 6)\).
04

Apply the Formula for Slope

The slope \( m \) is calculated using the formula \( m = (y_2 - y_1) / (x_2 - x_1) \). Using the points \((0, 0)\) and \((1, 6)\), we find the slope: \( m = (6 - 0) / (1 - 0) = 6 \).
05

Conclusion

The coordinates of the two points on the line are \((0, 0)\) and \((1, 6)\). The slope of the line is 6.

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Key Concepts

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

Coordinates of Points on a Line
To identify points that lie on a given line, such as the line defined by the equation \( y = 6x \), we need to find pairs of \( x \) and \( y \) values that satisfy the equation. These pairs are known as coordinates, represented as \((x, y)\). The process starts with selecting any values for \( x \), since the line is defined in terms of \( x \), and then substituting these into the equation to find corresponding \( y \) values. For example:
  • If we choose \( x = 0 \), substituting into the equation gives \( y = 6(0) = 0 \), resulting in the coordinate \((0,0)\).
  • Choosing \( x = 1 \), substituting gives \( y = 6(1) = 6 \), resulting in the coordinate \((1,6)\).
This method of choosing different \( x \) values helps us find multiple points, all of which lie on the line described by the equation. These points can then be plotted on a graph to visualize the line.
Equation of a Line
The equation of a line provides a mathematical description of the relationship between the \( x \) and \( y \) coordinates on a two-dimensional plane. In the equation \( y = 6x \), you'll notice it's in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The term \( 6x \) indicates the slope is 6, and since there's no \( + b \) term, \( b = 0 \), meaning the line intercepts the y-axis at 0.

This equation tells us several things:
  • The line passes through the origin point \((0, 0)\).
  • For every unit of increase in \( x \), \( y \) increases by 6 units, indicating a steep incline.
  • The line extends infinitely in both directions along its path defined by this linear relationship between \( x \) and \( y \).
Calculating Slope
Slope is a key concept in understanding lines and their behavior. It measures the steepness or incline of a line and is calculated as the ratio of the change in \( y \) to the change in \( x \) between two points on the line. Given two points \((x_1, y_1)\) and \((x_2, y_2)\), the formula for calculating slope \( m \) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
  • Using the points \((0, 0)\) and \((1, 6)\), substitute into the formula to get \( m = \frac{6 - 0}{1 - 0} = 6 \).
  • This tells us the slope of the line is 6, consistent with the coefficient of \( x \) in the line's equation \( y = 6x \).
  • A slope of 6 means for every one unit increase in \( x \), \( y \) increases by 6 units.
Understanding slope is fundamental for analyzing the direction and steepness of lines, which is essential when plotting graphs or interpreting linear relationships.

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Most popular questions from this chapter

For Problems \(23-32\), find the equation of the line with the given slope and \(y\) intercept. Leave your answers in slope-intercept form. $$ m=-2 \text { and } b=\frac{7}{3} $$

Write the equation of the line that satisfies the given conditions. Express final equations in standard form. \(x\) intercept of 5 and slope of \(-\frac{3}{10}\)

Write the equation of the line that satisfies the given conditions. Express final equations in standard form. Contains the origin and is perpendicular to the line \(-2 x+3 y=8\)

Find the equation of the line that contains the given point and has the given slope. Express equations in the form \(A x+B y=C\), where \(A, B\), and \(C\) are integers. (Objective 1a) $$(5,-6), m=\frac{3}{5}$$

Write the equation of the line that satisfies the given conditions. Express final equations in standard form. Contains the point \((5,6)\) and is perpendicular to the \(y\) axis
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