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Chapter 7: Problem 2

Write an equation in slope-intercept form for each line. slope \(=0, y\) -intercept \(=-7\)

Short Answer

Expert verified
The equation in slope-intercept form is \( y = -7 \).

Step by step solution

01

Understand 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.
02

Identify Given Values

From the problem, we know that the slope \( m = 0 \) and the y-intercept \( b = -7 \).
03

Substitute Values into Slope-intercept Form

Substitute the values for \( m \) and \( b \) into the equation \( y = mx + b \). This becomes \( y = 0x - 7 \).
04

Simplify the Equation

Since \( 0x = 0 \), the equation simplifies to \( y = -7 \). This is the equation of the line.

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Key Concepts

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

Linear Equations
Linear equations are a type of equation that forms a straight line when plotted on a graph. They are fundamental in algebra and represent relationships with constant rates of change. A linear equation can be written in various forms, but the most commonly used is the slope-intercept form. This is written as \( y = mx + b \), where:
  • \( y \) is the dependent variable (usually on the vertical axis).
  • \( x \) is the independent variable (usually on the horizontal axis).
In this form, \( m \) represents the slope and \( b \) indicates the y-intercept.
The slope tells you how steep the line is, and the y-intercept tells you where the line crosses the y-axis. Together, they give you a complete picture of the line's orientation and position.
Y-Intercept
The y-intercept is a key feature of a line in the slope-intercept form of a linear equation. It indicates the point where the line crosses the y-axis. Since the y-axis is where \( x = 0 \), the y-intercept is simply the value of \( y \) when \( x \) is zero. In the equation \( y = mx + b \):
  • \( b \) is the y-intercept.
This value is crucial as it defines the starting point of the line. Understanding the y-intercept helps in graphing the line and understanding its relationship with other points on the graph. In the original exercise, the y-intercept is given as \( -7 \), meaning the line crosses the y-axis at -7.
Slope
The slope of a line represents its steepness and direction. In the equation \( y = mx + b \), the slope \( m \) quantifies how much \( y \) changes for a unit change in \( x \). More specifically:
  • If \( m \) is positive, the line rises as you move from left to right.
  • If \( m \) is negative, the line falls as you move from left to right.
  • If \( m = 0 \), the line is horizontal, indicating no rise or fall.
In the exercise, \( m = 0 \), meaning the line is horizontal. This means it remains constant for any value of \( x \), leading to a flat line on the graph.
Equation Simplification
Equation simplification is a crucial step in solving for the most compact form of an equation. By simplifying, we express the equation in a way that's easy to understand and interpret. In the slope-intercept form \( y = mx + b \), if the slope \( m = 0 \),
  • Then the term \( 0x = 0 \).
This effectively removes \( x \) from the equation, simplifying it to \( y = b \). For the original exercise, what appears to be a complex equation \( y = 0x - 7 \) simplifies to \( y = -7 \). This concise form provides a clear understanding that regardless of \( x \), the y-value remains constant at -7. Simplifying equations like this helps in easy interpretation and graphing of lines.

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Most popular questions from this chapter

Subtract. $$-26-(-26)$$

Write each number in scientific notation. $$345,000$$

Make a scatter plot with at least ten points that appear to be somewhat linear. Draw two different lines that could approximate the data.

Carlotta and Alex are finding the slope and \(y\) -intercept of \(x+2 y=8 .\) Who is correct? Explain your reasoning. $$\begin{array}{ll}\text { Carlotta } & \text { Alex } \\ \text { slope }=2 & \text { slope } \sim-\frac{1}{2} \\ \text { y-intercept }=8 & y \text { -intercept }=4\end{array}$$

Graph each equation using the slope and \(y\) -intercept. $$y=2 x-3$$
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