How do you write an equation of a line passing through (4, 2), perpendicular to #y=2x+3#?

Answer 1

See the entire solution process below:

The equation in the problem is in slope-intercept form. The slope-intercept form of a linear equation is: #y = color(red)(m)x + color(blue)(b)#
Where #color(red)(m)# is the slope and #color(blue)(b)# is the y-intercept value.
#y = color(red)(2)x + color(blue)(3)#
Therefore the slope of this line is #m = 2#
Let us call the slope of the perpendicular line #m_p#. By definition, the slope of a perpendicular line is:
#m_p = -1/m#

Therefore, for this problem, the slope of the perpendicular line is:

#m_p = -1/2#
We can use the slope-intercept formula, substitute the values from the points for #x# and #y# and substitute the slope we determined and solve for #b#:
#2 = (color(red)(-1/2) * 4) + color(blue)(b)#
#2 = -2 + color(blue)(b)#
#color(red)(2) + 2 = color(red)(2) - 2 + color(blue)(b)#
#4 = 0 + color(blue)(b)#
#4 = color(blue)(b)#

Which means the equation is:

#y = color(red)(-1/2)x + color(blue)(4)#
Or, we could use the point-slope formula to also write and equation for the line. The point-slope formula states: #(y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))#
Where #color(blue)(m)# is the slope and #color(red)(((x_1, y_1)))# is a point the line passes through.

Substituting the slope we calculated and the values from the point in the problem gives:

#(y - color(red)(2)) = color(blue)(-1/2)(x - color(red)(4))#
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Answer 2

To write the equation of a line perpendicular to ( y = 2x + 3 ) passing through the point ( (4, 2) ), you need to determine the slope of the perpendicular line.

The slope of the given line ( y = 2x + 3 ) is 2.

Since perpendicular lines have slopes that are negative reciprocals of each other, the slope of the perpendicular line will be ( -\frac{1}{2} ).

Now that we have the slope ( -\frac{1}{2} ) and the point ( (4, 2) ), we can use the point-slope form of a linear equation:

[ y - y_1 = m(x - x_1) ]

Substitute the values:

[ y - 2 = -\frac{1}{2}(x - 4) ]

Now, expand and simplify:

[ y - 2 = -\frac{1}{2}x + 2 ]

[ y = -\frac{1}{2}x + 4 ]

So, the equation of the line passing through ( (4, 2) ) and perpendicular to ( y = 2x + 3 ) is ( y = -\frac{1}{2}x + 4 ).

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Answer from HIX Tutor

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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