Nolan plots the [tex]$y$[/tex]-intercept of a line at [tex]$(0,3)$[/tex] on the [tex]$y$[/tex]-axis. He uses a slope of 2 to graph another point. He draws a line through the two points. Which equation represents Nolan's line?
A. [tex]$y=2 x+1$[/tex]
B. [tex]$y=2 x+3$[/tex]
C. [tex]$y=3 x+2$[/tex]
D. [tex]$y=3 x+5$[/tex]
Answer (1)
The y-intercept is 3, meaning the line passes through the point (0,3).
The slope is 2, indicating that for every 1 unit increase in x, y increases by 2 units.
Using the slope-intercept form y = m x + b , substitute m = 2 and b = 3 .
The equation of the line is y = 2 x + 3 .
Explanation
Understanding the Problem We are given that the y-intercept of the line is at (0,3) and the slope of the line is 2. We need to find the equation of the line.
Using Slope-Intercept Form The slope-intercept form of a linear equation is given by y = m x + b , where m is the slope and b is the y-intercept.
Substituting Values We are given that the slope m = 2 and the y-intercept b = 3 . Substituting these values into the slope-intercept form, we get: y = 2 x + 3
Finding the Equation Therefore, the equation of the line is y = 2 x + 3 .
Examples
Imagine you're climbing a staircase. The y-intercept is where you start on the vertical axis (height), and the slope is how much your height increases with each step you take horizontally. If you start at a height of 3 feet and each step raises you 2 feet, the equation y = 2 x + 3 describes your height ( y ) after taking x steps. This concept is useful in various real-life situations, such as calculating the cost of a service based on a fixed initial fee and a per-unit charge, or determining the distance traveled by an object moving at a constant speed from an initial position.
