Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 4: Problem 9
Find the equation of a line that a.Has a positive slope and a negative \(y\)-intercept. b. Has a negative slope and a \(y\)-intercept of 0 . c. Passes through the points \((1,7)\) and \((4,10)\). d. Passes through the points \((-2,10)\) and \((4,10)\).
Short Answer
Expert verified
a. y = x - 3; b. y = -2x; c. y = x + 6; d. y = 10.
Step by step solution
01
Understanding Conditions for a Line Equation
To find the equation of a line, use the slope-intercept form: \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We'll apply this form under different conditions presented in the exercise.
02
Positive Slope and Negative Y-Intercept
Choose a positive value for the slope, such as \( m = 1 \). For the y-intercept, we need a negative value, say \( b = -3 \). Thus, the equation is \( y = x - 3 \).
03
Negative Slope and Y-Intercept 0
Select a negative slope, for example, \( m = -2 \). The y-intercept is \( b = 0 \) as specified. The equation of the line is \( y = -2x \).
04
Calculating Slope for Given Points (1,7) and (4,10)
To find the slope \( m \) for the points \((1,7)\) and \((4,10)\), use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 7}{4 - 1} = 1 \).
05
Writing the Equation for Points (1,7) and (4,10)
Using the point-slope form \( y - y_1 = m(x - x_1) \) with \( m = 1 \) and \( (x_1, y_1) = (1,7) \), the equation becomes \( y - 7 = 1(x - 1) \), or \( y = x + 6 \).
06
Identifying Slope for Horizontal Line Through (-2,10) and (4,10)
Since both points have the same y-coordinate, the slope is 0, indicating a horizontal line.
07
Equation for Horizontal Line
The equation of a line with a slope of 0 passing through y = 10 is simply \( y = 10 \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope-Intercept Form
The slope-intercept form of a line's equation is foundational in understanding how to express a line graphically. It is represented by the equation \( y = mx + b \). Here, \( m \) symbolizes the slope of the line, which measures the tilt or steepness. Meanwhile, \( b \) represents the \( y \)-intercept, where the line crosses the \( y \)-axis.
This form is particularly useful because it provides direct clues about the line's behavior:
This form is particularly useful because it provides direct clues about the line's behavior:
- A positive \( m \) indicates that the line rises as it moves from left to right.
- A negative \( m \) shows the line falls as it progresses from left to right.
- The value of \( b \) directly shows where the line intersects with the \( y \)-axis.
Slope Calculation
Calculating the slope of a line is crucial for describing how steep the line is. To find the slope between two points, such as \((x_1, y_1)\) and \((x_2, y_2)\), we use the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \].
This formula measures the change in y-values, also known as the "rise," over the change in x-values, or the "run."
This formula measures the change in y-values, also known as the "rise," over the change in x-values, or the "run."
- If the slope (\( m \)) is positive, the line goes up as you move left to right.
- If \( m \) is negative, the line goes down.
- A zero slope means the line is flat, or horizontal.
Horizontal Line
A horizontal line is unique because its slope is always zero. This occurs when there is no vertical change no matter how far we measure horizontally.
When coordinates of two points on a line share the same \( y \)-value, such as \((-2,10)\) and \((4,10)\), it indicates a horizontal line.
The equation for such a horizontal line is straightforward: \( y = \text{{constant}} \). For our example, it becomes \( y = 10 \).
Horizontal lines are easy to mistake for having no equation, but in reality, they reveal that the height of the line never changes鈥攑erfect for representing steady states or levels in various contexts.
When coordinates of two points on a line share the same \( y \)-value, such as \((-2,10)\) and \((4,10)\), it indicates a horizontal line.
The equation for such a horizontal line is straightforward: \( y = \text{{constant}} \). For our example, it becomes \( y = 10 \).
Horizontal lines are easy to mistake for having no equation, but in reality, they reveal that the height of the line never changes鈥攑erfect for representing steady states or levels in various contexts.
Point-Slope Form
The point-slope form of a line's equation is especially beneficial when you know a point on the line and the line's slope. It is shown as: \[ y - y_1 = m(x - x_1) \].
Here:
For example, if a line passes through the point \((1,7)\) with a slope of 1, substitute these into the formula:\( y - 7 = 1(x - 1) \), which simplifies to \( y = x + 6 \).
This form is perfect for finding equations of lines without immediately translating to slope-intercept form, particularly helpful in more advanced graphing situations or when data isn鈥檛 initially given in a direct line form.
Here:
- \( (x_1, y_1) \) is a known point on the line.
- \( m \) is the slope.
For example, if a line passes through the point \((1,7)\) with a slope of 1, substitute these into the formula:\( y - 7 = 1(x - 1) \), which simplifies to \( y = x + 6 \).
This form is perfect for finding equations of lines without immediately translating to slope-intercept form, particularly helpful in more advanced graphing situations or when data isn鈥檛 initially given in a direct line form.
