Discuss the steps necessary to carry out each activity. Describe briefly what each line looks like in relation to the original given line.

Answer these two questions briefly in your own words:

What does it mean for one line to be parallel to another?

What does it mean for one line to be perpendicular to another?

Incorporate the following five math vocabulary words into your discussion. Use bold font to emphasize the words in your writing (Do not write definitions for the words; use them appropriately in sentences describing your math work.):

Origin
Given an equation of a line, find equations for lines parallel or perpendicular to it going through specified points. Find the appropriate equations and points from the table below. Simplify your equations into slope-intercept form.

# 1 y=1/4x-2;(8,-1)
# 2 y=-4x-5; (0,-1)

Ordered pair

X- or y-intercept

Slope

Reciprocal

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Whattaya got? In your own words, that is.

To find equations for lines parallel or perpendicular to a given line, we need to follow a few steps. First, we need to identify the slope of the given line. The slope is represented by the coefficient of x in the equation of the line (y = mx + b), where m is the slope. For example, in the equation y = 1/4x - 2, the slope is 1/4. Similarly, in the equation y = -4x - 5, the slope is -4.

To find a line parallel to the given line, we need to use the same slope. The point through which the parallel line passes is given in the question. In this case, it is (8, -1). So, using the point-slope form of a line (y - y1 = m(x - x1)), where (x1, y1) are the given coordinates, we can substitute the given values to find the equation. Plugging in the slope of 1/4 and the point (8, -1), we get y - (-1) = 1/4(x - 8) and simplifying it gives us y = 1/4x + 1.

To find a line perpendicular to the given line, we need to find the negative reciprocal of the slope. So, the negative reciprocal of 1/4 is -4/1 (or simply -4). Again, using the point-slope form and substituting the values from the question, we get y - (-1) = -4(x - 0), which simplifies to y + 1 = -4x.

In summary, the equation for the line parallel to y = 1/4x - 2 through the point (8, -1) is y = 1/4x + 1, and the equation for the line perpendicular to y = -4x - 5 through the point (0, -1) is y + 1 = -4x.

In these equations, the slope represents the inclination or steepness of the line. The ordered pairs (8, -1) and (0, -1) represent specific points on these lines. The x-intercept is where the line crosses the x-axis, and the y-intercept is where the line crosses the y-axis. The origin (0, 0) is the point where the x-axis and y-axis intersect.