How do you find the perpendicular distance between the straight lines #y=2x+7# and #y=2x+2#?
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To find the perpendicular distance between two parallel lines, ( y = mx + c_1 ) and ( y = mx + c_2 ), where ( m ) is the slope and ( c_1 ) and ( c_2 ) are the y-intercepts, we can use the following formula:
[ \text{Distance} = \frac{|c_2 - c_1|}{\sqrt{1 + m^2}} ]
For the given lines ( y = 2x + 7 ) and ( y = 2x + 2 ), both lines have the same slope ( m = 2 ).
The y-intercepts are ( c_1 = 7 ) for the first line and ( c_2 = 2 ) for the second line.
Substituting these values into the formula:
[ \text{Distance} = \frac{|2 - 7|}{\sqrt{1 + 2^2}} ] [ \text{Distance} = \frac{5}{\sqrt{5}} ] [ \text{Distance} = \frac{5}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} ] [ \text{Distance} = \frac{5\sqrt{5}}{5} ] [ \text{Distance} = \sqrt{5} ]
So, the perpendicular distance between the lines ( y = 2x + 7 ) and ( y = 2x + 2 ) is ( \sqrt{5} ) units.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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