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Chapter 4: Problem 12

Determine if each pair of lines is parallel, perpendicular, or neither. $$\begin{array}{l}y-2 x=3 \\\x+2 y=3\end{array}$$

Short Answer

Expert verified
The given lines are perpendicular since their slopes are negative reciprocals of each other, with \(m_{1} = 2\) and \(m_{2} = -\frac{1}{2}\), satisfying the condition \(m_1 * m_2 = -1\).

Step by step solution

01

Put the given equations in the slope-intercept form \(y = mx + b\)

To put the given equations in the slope-intercept form, we need to solve for y. We have the following equations: $$\begin{array}{l}y-2 x=3 \\\x+2 y=3\end{array}$$ For the first equation: $$y = 2x + 3$$ For the second equation: $$2y = -x + 3 \Rightarrow y = -\frac{1}{2}x + \frac{3}{2}$$ Now, we have the following slope-intercept forms of the given equations: $$\begin{array}{l}y=2 x+3 \\\y=-\frac{1}{2}x+\frac{3}{2}\end{array}$$
02

Identify the slopes (m) of both lines

To identify the slopes, we can simply look at the coefficients of the x terms in the slope-intercept form. For the first line, we have: $$y = 2x+3$$ The slope of this line is equal to 2. For the second line, we have: $$y = -\frac{1}{2}x + \frac{3}{2}$$ The slope of the second line is equal to -1/2.
03

Compare the slopes

Now that we have the slopes of both lines, we can compare them to see their relationship. The slopes are: First line: m1 = 2 Second line: m2 = -1/2
04

Determine if the lines are parallel, perpendicular, or neither

Using the relationship between the slopes, we can determine if the lines are parallel, perpendicular, or neither. Parallel: The lines are parallel if their slopes are equal. m1 = m2 Perpendicular: The lines are perpendicular if their slopes are negative reciprocals of each other. m1 * m2 = -1 Neither: The lines are neither parallel nor perpendicular if their slopes do not meet either of these conditions. Comparing the slopes of our given lines, we can see that: - m1 鈮� m2 (The lines are not parallel) - m1 * m2 = 2 * (-1/2) = -1 (The lines are perpendicular) Since the product of the slopes is -1, we can conclude that the given lines are perpendicular to each other.

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Key Concepts

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

Slope-Intercept Form
Understanding the slope-intercept form is essential in analyzing and graphing linear equations. The standard expression for this form is \(y = mx + b\), where:
  • \(m\) represents the slope of the line, which shows the rate of change or the steepness of the line.
  • \(b\) is the y-intercept, which is the point where the line crosses the y-axis.
This form allows us to easily identify and compare the slopes and y-intercepts of different lines. When working with the slope-intercept form:
  • Solving for \(y\) will show the slope and y-intercept directly.
  • It simplifies the process of graphing the line on a coordinate plane.
In the original exercise, we converted both given equations into the slope-intercept form to determine their respective slopes and relationships.
Linear Equations
Linear equations are equations of the first degree, meaning they have no exponents higher than 1 and graph as straight lines on a coordinate plane. They can be expressed in various forms, with the slope-intercept being one of the most common. The general approach to solving linear equations involves:
  • Rearranging the equation to isolate one of the variables, typically \(y\).
  • Balancing both sides of the equation to maintain equality.
The goal is often to simplify the equation, making it easier to determine the slope and y-intercept, as seen in step 1 of the solution. Understanding linear equations is crucial for solving systems of equations, determining parallel or perpendicular lines, and optimizing solutions in various mathematical contexts.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operators (such as addition and multiplication) that represent a specific value. In the realm of linear equations, they form the basis of expressing relationships between variables. Tasks involving algebraic expressions often include:
  • Collecting like terms to simplify expressions.
  • Isolating variables by applying operations to both sides of an equation.
In the given problem, we used algebraic manipulation to transform equations into the slope-intercept form. This manipulation is a key skill in algebra, enabling one to influence the layout of equations to better suit specific needs, such as solving for a particular variable or comparing two equations.
Slope Comparison
Comparing slopes is crucial when determining the relationship between two lines. The slope of a line describes its steepness and direction:
  • Parallel lines have identical slopes, indicating they will never intersect.
  • Perpendicular lines have slopes that are negative reciprocals of each other, meaning their product is -1.
  • If slopes don鈥檛 fit either criterion, lines are neither parallel nor perpendicular.
In our solution, the slopes of the two lines were compared: one had a slope of 2, and the other -1/2. Multiplying these slopes yielded -1, confirming they are perpendicular. Slope comparison is a powerful tool for understanding geometric relationships and confirming line orientation.

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Most popular questions from this chapter

Jenelle earns \(\$ 7.50\) per hour at her part-time job. Her total earnings, \(E\) (in dollars), for working \(t\) hr can be defined by the function $$ E(t)=7.50 t $$ a) Find \(E(10),\) and explain what this means in the context of the problem. b) Find \(E(15),\) and explain what this means in the context of the problem. c) Find \(t\) so that \(E(t)=210,\) and explain what this means in the context of the problem.

Write the slope-intercept form of the equation of the line, if possible, given the following information. \(m=2\) and \(y\) -intercept \((0,-11)\)

Write the slope-intercept form of the equation of the line, if possible, given the following information. horizontal line containing \((0,-8)\)

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=\frac{5}{x}$$

Graph each function using the slope and \(y\) -intercept. $$f(x)=-x+5$$
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