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Chapter 1: Problem 20

Find an equation of the line: with \(m=7\), containing \((1,7)\)

Short Answer

Expert verified
The equation of the line is \( y = 7x \).

Step by step solution

01

Recall the Slope-Intercept Form

To find the equation of a line, start with the slope-intercept form of the equation of a line: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept.
02

Substitute the Slope

Given the slope \( m = 7 \), substitute this value into the slope-intercept form: \[ y = 7x + b \]
03

Substitute the Point

Use the given point \(1, 7\). Substitute \( x = 1 \) and \( y = 7 \) into the equation: \[ 7 = 7(1) + b \]
04

Solve for the Y-intercept

Solve the equation from Step 3 to find the y-intercept \( b \): \[ 7 = 7 + b \] Subtract 7 from both sides: \[ b = 0 \]
05

Write the Final Equation

Substitute \( b = 0 \) back into the slope-intercept form to write the final equation of the line: \[ y = 7x \]

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Key Concepts

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

slope-intercept form
To start, understanding the basic equation format is essential. The slope-intercept form of a line's equation is written as follows: \[ y = mx + b \] Let's break this down:
  • \( y \) represents the y-coordinate of a point on the line
  • \( x \) represents the x-coordinate
  • \( m \) is the slope of the line (how steep the line is)
  • \( b \) is the y-intercept (where the line crosses the y-axis)
By using this format, you can plug in values for \( m \) and \( b \) to create the specific equation for your line.
slope
The slope (represented by \( m \)) is a measure of how steep a line is. It's calculated by the following ratio: \[ m = \frac{\Delta y}{\Delta x} \] This means the slope is the change in the y-coordinate divided by the change in the x-coordinate.
For our exercise: The slope is given as \( m = 7 \), which indicates that for every 1 unit you move horizontally (in the x-direction), the line moves 7 units vertically (in the y-direction).
This tells you how rapidly the line ascends or descends.
If the slope was negative, the line would slope downwards.
y-intercept
Finally, let's talk about the y-intercept (represented by \( b \)). This is the point where the line crosses the y-axis. At this point, the x-coordinate is always 0.
To find the y-intercept, we use a point that the line passes through and substitute the x and y values of this point into the slope-intercept equation to solve for \( b \).
In our example, we used the point \((1, 7)\). Substituting \( x = 1 \) and \( y = 7 \) into \( y = 7x + b \), we got:
\[ 7 = 7(1) + b \] Solving this equation, we found that \( b = 0 \).
Therefore, for our equation, the y-intercept is 0, meaning the line crosses the y-axis at the origin.

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