Problem 53 - 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. $$\begin{array}{r}x-2 y=2 \\\2 x-4 y=3\end{array}$$

Step-by-Step Solution

Answer

The two given linear equations, when graphed, yield parallel lines due the equal slopes. Thus, the lines are not perpendicular.

Step 1: Convert to Slope-Intercept form

Convert each equation to slope-intercept form \(y = mx + b\). To do this, isolate \(y\) in both equations. For \(x - 2y = 2\), performing calculations shows \(y = (1/2)x - 1\). Similarly, for \(2x - 4y = 3\), calculations reveal \(y = (1/2)x - (3/4)\).

Step 2: Graph the Lines and Verify the Slope

Plot both lines on the same coordinate system. The slope of each line is \(1/2\), which is noted by the coefficient of \(x\) in both equations. Hence, the lines are parallel.

Step 3: Determine If Lines Are Parallel or Perpendicular

The slopes of both lines are equal, which means the lines are parallel. Perpendicular lines have slopes that are negative reciprocals, which is not the case here, therefore, the lines are not perpendicular.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of writing equations of straight lines. In this form, the equation is written as: \(y = mx + b\). Here, \(m\) represents the slope, or steepness, of the line, while \(b\) is the y-intercept, the point where the line crosses the y-axis.

To convert a linear equation to this form, you need to isolate \(y\) on one side of the equation. For instance, consider the equation \(x - 2y = 2\). By rearranging it, so \(y\) is alone, you get \(y = \frac{1}{2}x - 1\).

This process of conversion helps in easily identifying both the slope and the y-intercept of the line, making it simpler to graph the line on a coordinate plane.

Parallel and Perpendicular Lines

Parallel lines have equal slopes. If you have two lines \(y = m_1x + b_1\) and \(y = m_2x + b_2\), they are parallel if \(m_1 = m_2\). This means they will never meet, as they rise at the same rate.

In the given exercise, both lines have a slope of \(\frac{1}{2}\), indicating they are parallel. They will never intersect on the graph.

Perpendicular lines, on the other hand, have slopes that are negative reciprocals. This means if one line has a slope \(m\), the other should have a slope \(-\frac{1}{m}\) to be perpendicular. In this case, because the slopes are the same, the lines are not perpendicular.

Linear Equations

Linear equations represent straight lines on a graph. They have a constant rate of change and are typically expressed as \(ax + by = c\). In the slope-intercept form, it translates to \(y = mx + b\), which we discussed earlier.

Graphing a linear equation involves plotting the line based on its slope and y-intercept. The equation \(x - 2y = 2\) becomes \(y = \frac{1}{2}x - 1\) in slope-intercept form, showing a slope of \(\frac{1}{2}\) and a y-intercept at -1.

Understanding linear equations allows you to explore relationships between variables and predict future trends when graphed. This basic understanding is essential for tackling more complex algebraic concepts later on.