IMAT 2018 Q56 [Equation of a Line]

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)