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Chapter 2: Problem 265

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular: $$2 x-6 y=12$$ $$-x+3 y=1$$

Short Answer

Expert verified
The lines are parallel.

Step by step solution

01

Rearrange the First Equation into Slope-Intercept Form

Start with the equation \(2x - 6y = 12\). To convert this into the slope-intercept form \(y = mx + b\), solve for \(y\):1. Move \(2x\) to the other side: \(-6y = -2x + 12\)2. Divide every term by \(-6\): \(y = \frac{1}{3}x - 2\).The slope of the first line is \(m_1 = \frac{1}{3}\).
02

Rearrange the Second Equation into Slope-Intercept Form

Take the equation \(-x + 3y = 1\) and rearrange it into the form \(y = mx + b\):1. Move \(-x\) to the other side: \(3y = x + 1\).2. Divide every term by 3: \(y = \frac{1}{3}x + \frac{1}{3}\).The slope of the second line is \(m_2 = \frac{1}{3}\).
03

Determine the Relationship Between Slopes

With both lines in slope-intercept form, compare their slopes. The slope of the first line is \(m_1 = \frac{1}{3}\) and the slope of the second line is \(m_2 = \frac{1}{3}\). 1. **Parallel Lines:** If the slopes are equal \((m_1 = m_2)\).2. **Perpendicular Lines:** If the slopes are negative reciprocals \((m_1 \times m_2 = -1)\).Since \(m_1 = m_2 = \frac{1}{3}\), the lines are parallel.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
The slope-intercept form of a linear equation is expressed as \(y = mx + b\). This specific format is helpful because it immediately reveals the slope \(m\) and the y-intercept \(b\) of the line. The slope \(m\) represents how steep the line is, while the intercept \(b\) indicates where the line crosses the y-axis.
To convert an equation from standard form to slope-intercept form, you need to solve for \(y\). This often involves isolating \(y\) on one side of the equation. Start by moving the \(x\)-term to the other side, ensuring it changes sign, and then divide through by the coefficient of \(y\). This approach was applied to the equation \(2x - 6y = 12\), transforming it into \(y = \frac{1}{3}x - 2\).
Understanding and using the slope-intercept form allows you to quickly determine and visualize the behavior or trend of a line on a graph.
Slope Comparison
To determine if lines are parallel, perpendicular, or neither, comparing their slopes is crucial. Parallel lines have the same slope, meaning they run in the same direction and never intersect. Perpendicular lines, on the other hand, have slopes that are negative reciprocals of each other. If the product of two slopes is -1, the lines are perpendicular.
For the given equations, the equations were converted to the slope-intercept form, and their slopes were calculated. Both equations yielded a slope of \(\frac{1}{3}\).
  • If the slopes \(m_1 = m_2 = \frac{1}{3}\), the lines are parallel.
  • If \(m_1 \times m_2 = -1\), the lines would be perpendicular.
Since the slopes are equal and not negative reciprocals, the lines are parallel. Slope comparison is instrumental in determining the spatial relationship between lines.
Algebraic Manipulation
Algebraic manipulation is the process of rearranging and simplifying equations to make them more useful or easier to work with. In analyzing relationships between lines, algebraic manipulation enables the transformation of linear equations from their standard form to slope-intercept form.
This manipulation involves a step-by-step solution, first moving variables or terms across the equation to isolate specific elements. Continuing from the previous example, moving \(x\) terms and isolating \(y\) requires careful algebraic manipulation. Then, you divide all terms to showcase \(y\) as a function of \(x\).
Mastering algebraic manipulation is essential for solving complex equations across various math problems. It plays a critical role in understanding how mathematical theories apply in practical situations, such as determining parallel and perpendicular relationships in geometry.

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Most popular questions from this chapter

For the following exercises, consider this scenario: The median home values in subdivisions Pima Central and East Valley (adjusted for inflation) are shown in Table 2.14. Assume that the house values are changing linearly. $$\begin{array}{|c|c|c|}\hline \text { Year } & {\text { Pima Central }} & {\text { East valley }} \\ \hline 1970 & {32,000} & {120,250} \\ \hline 2010 & {85,000} & {150,000} \\ \hline\end{array}$$ In which subdivision have home values increased at a higher rate?

A car rental company offers two plans for renting a car. Plan \(A : \$ 25\) per day and \(\$ 0.10\) per mile Plan B: \(\$ 40\) per day with free unlimited mileage How many miles would you need to drive for plan B to save you money?

For the following exercises, consider the data in Table 2.18, which shows the percent of unemployed in a city of people 25 years or older who are college graduates is given below, by year. $$\begin{array}{|c|c|c|c|c|c|}\hline \text { Year } & {2000} & {2002} & {2005} & {2007} & {2010} \\ \hline \text { Percent Graduates } & {6.5} & {7.0} & {7.4} & {8.2} & {9.0} \\ \hline\end{array}$$ Determine whether the trend appears to be linear. If so, and assuming the trend continues, find a linear regression model to predict the percent of unemployed in a given year to three decimal places

For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs show the population and the year over the ten-year span (population, year) for specific recorded years: $$(3,600,2000) ;(4,000,2001) ;(4,700,2003) ;(6,000,2006)$$ What is the correlation coefficient for this model to three decimal places of accuracy?

For the following exercises, use the graph in Figure 2.59, showing the profit, \(y\) , in thousands of dollars, of a company in a given year, \(x\) , where \(x\) represents years since 1980. In 2004, a school population was 1250. By 2012 the population had dropped to 875. Assume the population is changing linearly. a. How much did the population drop between the year 2004 and 2012? b. What is the average population decline per year? c. Find an equation for the population, \(P\), of the school \(t\) years after 2004.
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