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Chapter 3: Problem 26

Find the slope and the \(y\) -intercept of each line whose equation is given. $$-4 x+y=7$$

Short Answer

Expert verified
Slope: 4, y-intercept: 7

Step by step solution

01

Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of a line is given by the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. Start by rewriting the given equation \(-4x + y = 7\) in this form.
02

Isolate y

To convert \(-4x + y = 7\) to slope-intercept form, isolate \(y\) on one side of the equation.Add \(4x\) to both sides:\[-4x + y + 4x = 7 + 4x\]Which simplifies to:\[y = 4x + 7\]
03

Identify the Slope and the y-Intercept

From the slope-intercept form \(y = 4x + 7\), identify the slope \(m\) and the \(y\)-intercept \(b\). Here, \(m = 4\) and \(b = 7\).

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Key Concepts

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

Understanding Slope
The slope of a line, often represented by the letter \( m \), measures the steepness and direction of the line. In the equation of a line in slope-intercept form, \( y = mx + b \), the slope \( m \) tells us how much \( y \) will change for each unit change in \( x \).
  • If the slope is positive, the line rises as it moves from left to right.
  • If the slope is negative, the line falls as it moves from left to right.
  • A larger absolute value of the slope means a steeper line.
  • A slope of zero means the line is horizontal.
To further illustrate, let's take the equation we worked with: \( -4x + y = 7 \). After rearranging it into slope-intercept form, we get \( y = 4x + 7 \). The slope here is \( 4 \), which means that for every 1 unit increase in \( x \), \( y \) increases by 4 units. This results in a fairly steep line that goes upwards from left to right.
The Role of the Y-Intercept
The y-intercept is the point where the line crosses the y-axis. It provides a starting point for graphing the line and is represented by the letter \( b \) in the slope-intercept form \( y = mx + b \).
  • The y-intercept occurs where \( x = 0 \).
  • In our example, the equation \( y = 4x + 7 \) tells us that the y-intercept \( b \) is \( 7 \).
This means that our line crosses the y-axis at the point \( (0, 7) \). Understanding the y-intercept is crucial for graphing linear equations because it gives you a point through which the line passes, simplifying the process of plotting the line on a coordinate grid.
Linear Equations Simplified
Linear equations represent straight lines when graphed and can be written in various forms, with the most common being the slope-intercept form, \( y = mx + b \).
  • The standard form is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants.
  • To convert to slope-intercept form, solve for \( y \).
In our example, the original equation was \( -4x + y = 7 \). This is in a different format but can be easily rearranged to the slope-intercept form by isolating \( y \).
  • Adding \( 4x \) to both sides gives us \( y = 4x + 7 \).
This transformation helps us see the slope and y-intercept clearly, making it easier to understand and graph the equation. By converting linear equations to the slope-intercept form, we gain immediate insight into the slope and y-intercept, which are key to understanding the behavior of the line.

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Most popular questions from this chapter

In Exercises \(101-108\), each model is of the form \(f(x)=m x+b .\) In each case, determine what \(m\) and \(b\) signify. Cost of a Movie Ticket. The average price \(P(t)\) in dollars, of a movie ticket is given by \(P(t)=0.21 t+5.43,\) where \(t\) is the number of years since \(2000 .\)

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Simplify. $$2-\left(3-2^{2}\right)+10 \div 2 \cdot 5$$
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