Question

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Answer 1

Perpendicular

Explanation:

In order to classify the provided lines as parallel, perpendicular, or having no particular relationship, it is essential to analyze their slopes.

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Recall that the slope-intercept form of a linear equation is:

`\boxed{\left\begin{array}{ccc}\text{\underline{Slope-Intercept Form:}}\\\\y=mx+b\end{array}\right }`

Where 'm' represents the slope of the line.

• If two lines share the same slope, they are parallel.
• If the slopes of the lines are negative reciprocals of each other, the lines are perpendicular.
• When the slopes are neither equal nor negative reciprocals, the lines do not fall into either the parallel or perpendicular category.

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Our given set of equations:

8x - 2y = 3

x + 4y = -1

Let's rearrange the given equations into the slope-intercept form:

(1) 8x - 2y = 3

To get it into slope-intercept form, we solve for 'y':

⇒ -2y = -8x + 3

∴ y = 4x - 3/2

(2) x + 4y = -1

Solving for 'y':

⇒4y = -x - 1

∴y = (-1/4)x - 1/4

Now that we have the equations in slope-intercept form, we can see the slopes more clearly:

The slope of the first line is 4.

The slope of the second line is -1/4.

Since these slopes are negative reciprocals of each other, the lines are perpendicular.

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Additional Information:

Reciprocal: The reciprocal of a number is obtained by dividing 1 by that number. For a non-zero value 'a', its reciprocal 'b' is expressed as:

• b = 1 / a

Negative Reciprocal: The negative reciprocal of a number is formed by first finding its reciprocal and then negating the result (making it negative). For a non-zero number 'a', its negative reciprocal 'c' can be defined as:

• c = -1 / a