Problem 56 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(47-56,\) graph both linear equations in the same rectangular coordinate system. If the lines are parallel or perpendicular, explain why. $$\begin{array}{r} 3 x-y=-2 \\ x+3 y=-9 \end{array}$$

Step-by-Step Solution

Answer

The lines represented by the equations \(y = 3x + 2\) and \(y = -\frac{1}{3}x - 3\) are neither parallel nor perpendicular.

Step 1: Convert equations to slope-intercept form (y=mx+b)

For the first equation, solve for y: \(3x - y = -2\) rearranges to \(y = 3x + 2\). For the second equation, solve for y: \(x + 3y = -9\) rearranges to \(y = -\frac{1}{3}x - 3\).

Step 2: Compare the slopes

In the first equation, the slope (m1) is 3 and in the second equation, the slope (m2) is -1/3. Since m1 is not equal to m2 and m1*m2 is not equal to -1, the lines are neither parallel nor perpendicular.

Step 3: Graph the equations

Now graph the two equations on the same coordinate system. Use the y-intercept (b) to mark the first point, then use the slope (m) to mark another point. Draw a straight line through these points.

Key Concepts

Slope-intercept form

The slope-intercept form of a linear equation is one of the most recognizable forms when working with linear equations. This form is represented as \( y = mx + b \), where:

• \( y \) is the dependent variable or the value we're trying to find.
• \( m \) is the slope, which tells us how steep the line is.
• \( x \) is the independent variable.
• \( b \) is the y-intercept, showing where the line crosses the y-axis.

The slope-intercept form is helpful because it gives a direct insight into the line's behavior. The slope \( m \) indicates how much \( y \) changes when \( x \) increases by one unit.
For example, a slope of 3 means that for every increase of 1 in \( x \), \( y \) increases by 3 units.

To convert a standard linear equation to this form, rearrange the terms to isolate \( y \) on one side of the equation. This makes it easy to identify both the slope and intercept, helping us quickly graph the line or compare slopes between lines.

Graphing linear equations

Graphing linear equations involves plotting points on an x-y coordinate plane and drawing a straight line through them. To do this, you'll first need the equation in the slope-intercept form, \( y = mx + b \). The steps are simple:

• Identify the y-intercept \( b \) and plot it on the y-axis.
• From this point, use the slope \( m \) to find another point. If \( m \) is a fraction \( \frac{a}{b} \), move \( a \) units up (or down if negative) and \( b \) units right (or left if negative).
• Once you have two points, draw a line through them extending it across the graph.

This creates a visual representation of the linear equation. Observation of where lines intersect or if they are parallel or perpendicular can be easily seen once they are graphed.

Graphing helps us better understand the relationships within equations and offers a clear view of how changes in equations affect their graphical representation.

Parallel and perpendicular lines

Understanding when lines are parallel or perpendicular involves examining their slopes. For two lines to be parallel, they need to have the same slope, which means they rise and run at the same rate and will never intersect. In mathematical terms, if two lines have slopes \( m_1 \) and \( m_2 \), they are parallel if \( m_1 = m_2 \).

On the other hand, perpendicular lines intersect at a right angle. Their slopes are negative reciprocals of each other. This means if one line has a slope \( m_1 \), the perpendicular line will have a slope \( m_2 = -\frac{1}{m_1} \). For instance, if one slope is 2, the perpendicular slope is \( -\frac{1}{2} \).

In the given exercise, the slopes of the lines were 3 and -1/3. These specific slopes mean that the lines are neither parallel nor perpendicular, as their slopes do not satisfy the conditions above. This fundamental understanding of slopes ensures an easier analysis of lines and their relationships.