A. Without graphing, how can you tell that the graphs of y=2...

a. Without graphing, how can you tell that the graphs of $y=2 x+1$ and $y=3 x+2$ intersect? b. Without graphing, how can you tell that the graphs of $y=2 x+1$ and $y=2 x+2$ do not intersect? c. Without graphing, how can you tell that the graphs of $y=2 x+3$ and $2 y=4 x+6$ are the same line?

a. Without graphing, how can you tell that the graphs of $y=2 x+1$ and $y=3 x+2$ intersect?
b. Without graphing, how can you tell that the graphs of $y=2 x+1$ and $y=2 x+2$ do not intersect?
c. Without graphing, how can you tell that the graphs of $y=2 x+3$ and $2 y=4 x+6$ are the same line?

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Linear Equations in Slope-Intercept Form

Linear equations expressed in slope-intercept form (y = mx + b) clearly show two fundamental characteristics: the slope (m) and the y?intercept (b). This form provides a straightforward way to compare different lines, as the slope indicates the line’s steepness and direction while the y-intercept serves as the point where the line crosses the y-axis.

Comparison of Slopes

The slope of a line determines how steeply it ascends or descends. When comparing two linear equations, if the slopes are different, the lines must intersect at one unique point. Conversely, if the slopes are equal, the lines are either parallel (if their y-intercepts differ) or coincident (if their y-intercepts are also equal), which means they either never intersect or are the same line.

Parallel and Coincident Lines

Parallel lines have identical slopes but different y-intercepts, meaning they never meet since they run side-by-side at a constant distance apart. In contrast, coincident lines occur when two equations, despite having different appearances, are algebraically equivalent, meaning they represent the same set of points on the plane. This algebraic equivalence confirms that the graphs are, in fact, the same line.

Transcript

Please give Ace some feedback