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Question: How do you determine if \(5y+3x=1\) is parallel, perpendicular to neither to the line \(y+10x=-3\) ?...

How do you determine if 5y+3x=15y+3x=1 is parallel, perpendicular to neither to the line y+10x=−3y+10x=-3 ?

Explanation

Solution

Firstly, here we find the slope for both the given linear line equations. Let us then name it m1,m2{{m}_{1}},{{m}_{2}} . To find if they are parallel to each other, then check for the condition, m1=m2{{m}_{1}}={{m}_{2}} . To check if they are perpendicular to each other, then check for the condition, m1m2=−1{{m}_{1}}{{m}_{2}}=-1 .By checking out these two conditions we shall know if both the lines are parallel or perpendicular or neither.

Complete step by step answer:
The given line equations are, 5y+3x=1;y+10x=−35y+3x=1;y+10x=-3
To check if a pair of lines are parallel or perpendicular or neither we must first find their slopes.
Any straight line can be written in slope-intercept form, y=mx+cy=mx+c
Where mm is said to be the slope of the line (m=tan⁡θ)\left( m=\tan \theta \right)
And cc is the y-intercept.
Now writing the first equation in slope-intercept form to find the slope.
⇒5y+3x=1\Rightarrow 5y+3x=1
⇒5y=1−3x\Rightarrow 5y=1-3x
Now isolate the yy variable.
⇒y=1−3x5\Rightarrow y=\dfrac{1-3x}{5}
Now separate the terms.
⇒y=15−3x5\Rightarrow y=\dfrac{1}{5}-\dfrac{3x}{5}
Comparing it with the slope-equation, y=mx+by=mx+b
⇒m1=−35;b=15\Rightarrow {{m}_{1}}=\dfrac{-3}{5};b=\dfrac{1}{5}
Now writing the second equation in slope-intercept form to find the slope.
⇒y+10x=−3\Rightarrow y+10x=-3
Now isolate the yy variable.
⇒y=−3−10x\Rightarrow y=-3-10x
Comparing it with the slope-equation, y=mx+by=mx+b
⇒m2=−10;b=−3\Rightarrow {{m}_{2}}=-10;b=-3
Now To check if a pair of lines are parallel, their slopes must be equal m1=m2{{m}_{1}}={{m}_{2}}
Here m1=−35;m2=−10{{m}_{1}}=\dfrac{-3}{5};{{m}_{2}}=-10
Since they are not equal, they are not parallel.
Now to check if a pair of lines are perpendicular, they should satisfy the condition, m1m2=−1{{m}_{1}}{{m}_{2}}=-1
m1=−35;m2=−10{{m}_{1}}=\dfrac{-3}{5};{{m}_{2}}=-10
⇒m1m2=−35×−10=6\Rightarrow {{m}_{1}}{{m}_{2}}=\dfrac{-3}{5}\times -10=6
⇒6≠1\Rightarrow 6\ne 1
Since the slopes did not satisfy the condition, they are not perpendicular.
Hence the lines are not parallel and not perpendicular.

Note:
The slope of a line is the steepness of a line in a horizontal or vertical direction. The slope of a line can be calculated by taking the ratio of the change in vertical dimensions upon the change in horizontal dimensions which is given by,
The formula for finding slope when two points are given is,m=(y2−y1)(x2−x1)m=\dfrac{\left( {{y}_{2}}-{{y}_{1}} \right)}{\left( {{x}_{2}}-{{x}_{1}} \right)}