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Chapter 7: Problem 90

Determine whether each pair of lines is parallel, perpendicular, or neither $$ 2 x+y=6 \text { and } x-y=4 $$

Short Answer

Expert verified
Neither

Step by step solution

01

Convert Equations to Slope-Intercept Form

The slope-intercept form of a line is given by the equation: \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept. First, convert the first equation \( 2x + y = 6 \) to this form by solving for \( y \).
02

Solve the First Equation

Start by isolating \( y \): \( y = -2x + 6 \). This shows that the slope \( m_1 \) of the first line is -2.
03

Convert the Second Equation to Slope-Intercept Form

Next, convert the second equation \( x - y = 4 \) into slope-intercept form by solving for \( y \).
04

Solve the Second Equation

Isolate \( y \): \( y = x - 4 \). This shows that the slope \( m_2 \) of the second line is 1.
05

Compare the Slopes

Recall that: - Lines are parallel if and only if their slopes \( m \) are equal.- Lines are perpendicular if the product of their slopes is -1.- If neither condition is met, the lines are neither parallel nor perpendicular.Here, \( m_1 = -2 \) and \( m_2 = 1 \). Since \( -2 e 1 \) and \( -2 \times 1 = -2 eq -1 \), the lines are neither parallel nor perpendicular.

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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 is a simple and common way to write the equation of a line. It is written as: \[ y = mx + b \]
  • \( y \) is the dependent variable or the value of the function at any point.
  • \( m \) is the slope of the line. It indicates how steep the line is and its direction鈥攑ositive for upward, negative for downward.
  • \( x \) is the independent variable.
  • \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
For example, converting the equation \( 2x + y = 6 \) to slope-intercept form involves isolating \( y \). Subtract \( 2x \) from both sides to get: \[ y = -2x + 6 \]. Now the equation is in slope-intercept form with a slope (\( m \)) of -2 and a y-intercept (\( b \)) of 6. This process helps in visualizing the relationship between \( x \) and \( y \).
Solving Linear Equations
Solving linear equations often requires manipulating the equation to isolate the variable of interest. Let's look at the second equation from our problem: \( x - y = 4 \). To convert this into slope-intercept form, we need to solve for \( y \). Start by subtracting \( x \) from both sides: \[ -y = -x + 4 \]. Next, multiply every term by -1 to solve for \( y \): \[ y = x - 4 \].
  • The slope \( m \) here is 1.
  • The y-intercept \( b \) is -4.
By converting linear equations into slope-intercept form, we make it easier to analyze the lines described by those equations and understand the relationships between their slopes and y-intercepts.
Slope Comparison
Comparing slopes is essential to determine if lines are parallel, perpendicular, or neither. Here are the basics:
  • Lines are parallel if their slopes \( m \) are the same.
  • Lines are perpendicular if the product of their slopes is -1.
  • If neither condition is met, the lines are neither parallel nor perpendicular.
In our exercise, we found the slopes of the lines to be \( -2 \) and \( 1 \). Compare them:
1. \( m_1 = -2 \) and \( m_2 = 1 \) are not equal, so the lines are not parallel.
2. The product of their slopes is \( -2 \times 1 = -2 \), which is not equal to -1. Thus, they are not perpendicular.
Therefore, the lines represented by the equations \( 2x + y = 6 \) and \( x - y = 4 \) are neither parallel nor perpendicular. Understanding slope comparison helps in solving similar problems and validating the relationships between different lines.

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Most popular questions from this chapter

The average price of a movie ticket in 2004 was \(\$ 6.21 .\) In \(2016,\) the average price was \$8.65. Find and interpret the average rate of change in the price of a movie ticket per year to the nearest cent. (Data from Motion Picture Association of America.)

Solve each system by the elimination method. Check each solution. $$ \begin{array}{l} 7 x+2 y=0 \\ 4 y=-14 x \end{array} $$

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Without graphing, do the following for each system of equations. (a) Describe each system. (b) State the number of solutions. (c) Is the system inconsistent, are the equations dependent, or neither? \(y+2 x=6\) \(x-3 y=-4\)

Graph each line passing through the given point and having the given slope $$ (2,-5) ; m=0 $$
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