Which one of the following is an equation of the line that passes through (4, 3) and is perpendicular to the line y = 2x + 4?
A. 2x – y = 5
B. 2y + x = 11
C. 2y – x = 2
D. 2x + y = 11
E. 2y + x = 10
The basic equation of a line is y=kx+n.
Knowing that the line passes through (4,3) which is (x_1,y_1) we can put the information in the basic equation:
y_1=kx_1 +n \to 3= 4k +n \to n= 3-4k
Knowing that the if the line is perpendicular to the line y = 2x + 4. Perpendicular lines are lines that intersect at right angles . If you multiply the slopes of two perpendicular lines in the plane, you get −1 . That is, the slopes of perpendicular lines are opposite reciprocals .
The slopes are the k in the basic equation. So in this case, the slope needed is: k = \frac{-1}{2} = -\frac{1}{2}
So, using that information in the equation for n: n= 3-4k \to n= 3 - 4 \cdot (-\frac{1}{2})= 3+2 = 5
Integrating all of the data in the original form:
y = -\frac{1}{2} \cdot x +5 \to \text{multiplying ut by 2} \to 2y = -x +10 \to 2y + x = 10 .
So the final correct answer is (E)
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But how is the slope of one line equall to 2. Clarify please!
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y = mx+ c ; y = 2x + 4. Therefore m (gradient) of the given line is 2