Let's solve the first equation for y.
5x-6y = 36
-6y = 36-5x
-6y = -5x+36
y = (-5x+36)/(-6)
y = (-5x)/(-6)+36/(-6)
y = (5/6)x - 6
This equation is in slope-intercept form y = mx+b where,
- m = 5/6 = slope
- b = -6 = y intercept
The slope is all we care about when looking for parallel or perpendicular lines.
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Now we need to find the slope of the other equation.
Let's solve for y.
x+5y = 4
5y = 4-x
5y = -x+4
y = (-x+4)/5
y = (-x/5)+4/5
y = (-1/5)x + 4/5
The slope here is -1/5
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Recap so far
- Slope of 5x-6y=36 is 5/6
- Slope of x+5y=4 is -1/5
The slopes aren't equal, so the lines aren't parallel. Parallel lines occur when the slopes are the same but the y intercepts are different.
The slopes aren't perpendicular because (5/6)*(-1/5) = -1/6 isn't equal to -1. Perpendicular slopes multiply to -1 assuming neither line is vertical nor horizontal.
In other words, 5/6 and -1/5 are not negative reciprocals of one another to prove the lines aren't perpendicular.
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Final Answer: Neither