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Chapter 3: Problem 14

Write an equation in slope-intercept form of the line having the given slope and \(y\) -intercept. $$m: 2,(0,8)$$

Short Answer

Expert verified
The equation is \( y = 2x + 8 \).

Step by step solution

01

Understand the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) represents the \( y \)-intercept.
02

Identify the Values

From the problem statement, we know that the slope \( m = 2 \) and the \( y \)-intercept is \( 8 \).
03

Substitute Values into the Equation

Now, substitute the given values into the slope-intercept form. Here, \( m = 2 \) and \( b = 8 \), so the equation becomes \( y = 2x + 8 \).

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Key Concepts

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

Linear Equations
Linear equations form the backbone of algebra. A linear equation creates a straight line when plotted on a graph. It's called "linear" because it describes a line, and it is typically expressed in terms of variables and coefficients.
  • Linear equations are essential in representing relationships between two variables that change linearly.
  • Any linear equation can be expressed in the slope-intercept form, which is a convenient format.
  • The general form of a linear equation is written as \( y = mx + b \).
This equation graphically represents a straight line where each term plays an important role. The "\( y \)" stands for the dependent variable, while "\( x \)" is the independent variable. It's a direct relationship, easily visualized on a graph with a predictable pattern.
Slope
The slope of a line in mathematics is a number that describes both the direction and steepness of the line. In simpler terms, it tells us how much \( y \) changes for a given change in \( x \).
  • The slope is usually symbolized by the letter \( m \) in the equation \( y = mx + b \).
  • In the given example, \( m = 2 \), which means for every 1 unit increase in \( x \), \( y \) increases by 2 units.
A positive slope indicates an upward trend, whereas a negative slope indicates a downward slope.
  • Zero slope: The line is horizontal, showing no change in \( y \) as \( x \) changes.
  • Undefined slope: The line is vertical, meaning \( x \) doesn't change as \( y \) changes; hence, the slope is not defined.
Understanding the slope is crucial as it shows the angle of the line and essentially defines the behavior of linear functions.
Y-intercept
The \( y \)-intercept is a key component in the equation of a line, acting as the starting point where the line crosses the \( y \)-axis.
  • In the slope-intercept form \( y = mx + b \), \( b \) represents the \( y \)-intercept.
  • It indicates the value of \( y \) when \( x = 0 \).
In our given example, the \( y \)-intercept is \( 8 \). This means the line crosses the \( y \)-axis at the point \((0, 8)\). This figure is significant because it provides a fixed point for the line on the graph when no other coordinates are given.
  • Having the \( y \)-intercept allows us to quickly graph the line by starting at this point and following the slope from there.
The \( y \)-intercept, when combined with the slope, equips you with all necessary information to graphically plot your linear equation.

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Most popular questions from this chapter

Write an equation in slope-intercept form of the line having the given slope and \(y\) -intercept. $$m: \frac{5}{8},(0,-6)$$

Give a real-world example of a line with a slope of 0 and a real-world example of a line with an undefined slope.

Write an equation in point-slope form of the line with the given slope that contains the given point. $$m=-5,(4,7)$$

Simplify each expression. $$\frac{2}{9} \cdot\left(-\frac{18}{5}\right)$$

Simplify each expression. $$\frac{14-11}{23-15}$$
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