Past Question and Solution
Obafemi Awolowo University, Ile-Ife, Osun, Nigeria
MTH 106: Elementary Mathematics
Topic: Equations of Parallel Lines.
Question:
Find the equation of the line passing through the origin and parallel to the line 3๐ฅ + 2๐ฆ + 4 = 0.
Solution:
First, re-write the given equation in the slope-intercept form, ๐ฆ = ๐๐ฅ + ๐, where:
๐ = slope of the line
๐ = ๐ฆ โ intercept of the line
3๐ฅ + 2๐ฆ + 4 = 0
Subtract 3๐ฅ and 4 to both sides of the equation to isolate 2๐ฆ on the left-hand side:
3๐ฅ โ 3๐ฅ + 2๐ฆ + 4 โ 4 = 0 โ 3๐ฅ โ 4
The additive inverses become zero:
0 + 2๐ฆ + 0 = 0 โ 3๐ฅ โ 4
Remove the zeros:
2๐ฆ = โ3๐ฅ โ 4
Divide through by 2 to isolate ๐ฆ on the left hand-side:
2๐ฆ
2= โ 3๐ฅ
2โ4
2
Simplify the equation:
๐ฆ = โ 3
2๐ฅ โ 2
So, the given equation is ๐ฆ = โ 3
2๐ฅ โ 2 in the slope-intercept form.
The slope is โ3
2
Note: Parallel lines have equal slopes.
Therefore, the slope of the required line is also โ3
2.
It is given that the line passes through the origin (0,0).
Now, put the point (0,0) and the slope โ3
2 into the point-slope form of a line ๐ฆ โ ๐ฆ1= ๐(๐ฅ โ ๐ฅ1),
where (๐ฅ1, ๐ฆ1) is a point on the line and ๐ is the slope:
๐ฆ โ 0 = โ 3
2(๐ฅ โ 0)
Rewrite the equation:
The required equation is ๐ฆ = โ 3
2๐ฅ
