Problem 52 - Introductory Algebra for College Students

Question

Graph both linear equations in the same rectangular coordinate system. If the lines are parallel or perpendicular, explain why. $$\begin{aligned}&y=x+2\\\&y=-x-1\end{aligned}$$

Step-by-Step Solution

Answer

The graphs of the two equations intersect at a right angle, so the lines are perpendicular. The lines would have been parallel if they had the same slope.

Step 1: Extract the Slope and the Y-intercept for Each Equation

For the equation \(y = mx + b\), 'm' stands for the slope and 'b' for the y-intercept. Therefore, for the first equation \(y = x + 2\), the slope is 1 and the y-intercept is 2. In the second equation \(y = -x - 1\), the slope is -1 and the y-intercept is -1.

Step 2: Draw the Graphs of Both Equations

Start by drawing a rectangular coordinate system. Plot the point corresponding to the y-intercept for each equation then use the slope (rise over run) to draw the line. For the first equation, start at (0,2) and, since the slope is 1 (which is 1/1), go up one unit and right one unit to draw the next point. Draw the line through these points. For the second equation, start at (0,-1) and, since the slope is -1 (which is -1/1), go down one unit and right one unit to draw the next point. Draw the line through these points.

Step 3: Determine If the Lines Are Parallel or Perpendicular

Parallel lines have the same slope, while perpendicular lines have negative reciprocal slopes. In this case, the slopes are 1 and -1, which are negative reciprocals of each other, so these lines are perpendicular.

Key Concepts

Slope

The concept of slope is fundamental when working with linear equations. Slope represents the steepness or incline of a line, illustrating how much the y-value changes for every unit increase in the x-value.
When you have a line described by the equation \( y = mx + b \), the slope is denoted by \( m \). In simple terms, the slope is the 'rise over run,' indicating vertically how much a line climbs or descends for every unit it progresses horizontally.

• If the slope is positive, the line ascends from left to right.
• If the slope is negative, the line descends from left to right.
• A zero slope indicates a horizontal line.
• An undefined slope corresponds to a vertical line.

For example, in the equation \( y = x + 2 \), the slope \( m \) is 1. This indicates for every one unit increase in \( x \), \( y \) also increases by one unit. In contrast, \( y = -x - 1 \) has a slope of \(-1\), showing that for every one unit increase in \( x \), \( y \) decreases by one unit.

Y-intercept

Understanding the y-intercept in a linear equation is critical when graphing or interpreting lines. The y-intercept is the point where the line crosses the y-axis. In the equation of the form \( y = mx + b \), the \( b \) represents the y-intercept.
The y-intercept tells us the initial value of \( y \) when \( x \) is 0. It is a key starting point for graphing. Knowing the y-intercept allows you to place a point accurately at \( (0, b) \) on the graph.
For instance, with \( y = x + 2 \), the y-intercept is 2, meaning the line crosses the y-axis at (0,2). Meanwhile, \( y = -x - 1 \) has a y-intercept of -1, indicating the line crosses at (0,-1). These intercepts give us precise initial points to start drawing our lines on the coordinate plane.

Perpendicular Lines

When discussing perpendicular lines, it's essential to examine the relationship between their slopes. Perpendicular lines intersect to form a right angle (90 degrees), and the product of their slopes is always -1. This property is because their slopes are negative reciprocals of each other.
For two lines described by \( m_1 \) and \( m_2 \), if \( m_1 \times m_2 = -1 \), the lines are perpendicular.
Consider the equations \( y = x + 2 \) and \( y = -x - 1 \), which have slopes 1 and -1, respectively. Here, \( 1 \times (-1) = -1 \), confirming that these lines are perpendicular. This means on a graph, they intersect creating angles of 90 degrees, a hallmark of perpendicularity. Understanding this property assists not only in mapping equations but also in solving geometric problems involving lines.