Problem 42 - Intermediate Algebra

Question

For the exercises below, see the Concept Check in this section. Without solving, how can you tell that the graphs of \(y=2 x+3\) and \(y=2 x+7\) do not have any points of intersection?

Step-by-Step Solution

Answer

The lines are parallel with the same slope and different y-intercepts, so they never intersect.

Step 1: Identify the Functions

Consider the two linear functions given: \(y = 2x + 3\) and \(y = 2x + 7\). Both are linear equations with the same slope.

Step 2: Understand the Slope

The slope of each line is indicated by the coefficient of \(x\), which is 2 for both equations. This means both lines have the same slope, making them parallel.

Step 3: Determine Parallelism

To determine whether two lines are parallel, examine their slopes. If their slopes are equal and their y-intercepts differ, they do not intersect. Here, the slopes are equal (both 2), and the y-intercepts are different: 3 and 7, respectively.

Step 4: Draw Conclusion from the Slopes and Intercepts

Since the lines are parallel (same slope) and not overlapping (different y-intercepts), they do not intersect. Parallel lines with different y-intercepts will never meet on a coordinate plane.

Key Concepts

Parallel Lines

Parallel lines are lines in the same plane that never intersect, no matter how far they are extended. This is because they have exactly the same slope. Imagine two railway tracks running beside each other into the distance. They remain the same distance apart at all points.

• Lines are parallel if they have the same slope but different y-intercepts.
• If they have the same slope and the same y-intercept, they are not just parallel; they are actually the same line.

Understanding parallel lines helps in predicting line behavior on a graph, especially how they relate to each other in terms of their positions on the coordinate plane.

Slope

The slope of a line indicates how steep it is and the direction it is going. It's often represented as the letter 'm' and is calculated by the change in y over the change in x (rise over run).

• The slope can be positive, negative, zero, or undefined.
• A positive slope means the line rises from left to right.
• A negative slope means the line falls from left to right.
• A zero slope means the line is horizontal, while an undefined slope means the line is vertical.

In equations like \(y = mx + b\), the number next to the \(x\) is the slope, helping to determine if lines are parallel or intersect at some point.

Y-intercept

The y-intercept is the point where the line crosses the y-axis of a graph. It is represented by the letter \(b\) in the equation \(y = mx + b\). For a linear equation, the y-intercept is the value of \(y\) when \(x\) is zero.

• It tells us where the line starts on the vertical axis.
• If two lines have the same slope, but different y-intercepts, they are parallel.

In the case of our specific equations \(y = 2x + 3\) and \(y = 2x + 7\), their y-intercepts are 3 and 7, explaining why they are parallel but not coincident.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can graph points, lines, and curves using two axes: the x-axis (horizontal) and the y-axis (vertical). Each point on this plane is identified by an ordered pair \((x, y)\).

• It helps visualize equations and how different graphs intersect or run parallel.
• Allows examination of how changes to equations affect the graph visually, such as shifts when altering the y-intercept.

In analyzing linear equations, the coordinate plane helps us see patterns, like parallel lines that have the same slopes but differ by shifts along the y-axis.