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i is the line 3x + 2y + 18 = 0 - Leaving Cert Mathematics - Question Question 1 - 2013
Question Question 1
i is the line 3x + 2y + 18 = 0. Find the slope of i. 3x + 2y + 18 = 0 => 2y = -3x - 18 => y = - rac{3}{2}x - 9 Slope = - rac{3}{2} The line k is perpendicula... show full transcript
Worked Solution & Example Answer:i is the line 3x + 2y + 18 = 0 - Leaving Cert Mathematics - Question Question 1 - 2013
Step 1
Find the slope of i.
Answer
To determine the slope of the line given by the equation , we first rearrange it into the slope-intercept form (y = mx + b).
Starting with:
We isolate y:
-
Subtract from both sides:
-
Divide everything by 2: y = - rac{3}{2}x - 9
From this equation, we can identify the slope as - rac{3}{2}.
Step 2
Find the equation of k.
Answer
The line k is perpendicular to line i at the point (7, 0). First, we need to determine the slope of k, which is the negative reciprocal of the slope of line i.
Since the slope of line i is - rac{3}{2}, the slope of line k is:
m_k = rac{2}{3}
To write the equation of line k using point-slope form, we start from:
Substituting , , and m = rac{2}{3}:
y - 0 = rac{2}{3}(x - 7)
This simplifies to:
ightarrow 2x - 3y - 14 = 0$$Step 3
Find the co-ordinates of the point of intersection of the lines i and k.
Answer
To find the intersection of the lines i and k, we equate the two equations:
-
From i:
-
From k:
Which simplifies to
We can solve these equations simultaneously.
From equation (1), express :
ightarrow y = 9 - rac{3}{2}x$$ Substituting this into the second equation: $$2x - 3(9 - rac{3}{2}x) = 14$$ Expanding and simplifying: $$2x - 27 + rac{9}{2}x = 14 ightarrow rac{13}{2}x - 27 = 14$$ Rearranging gives: $$ rac{13}{2}x = 41 ightarrow x = -2$$ Now substituting $x = -2$ back into one of the equations to find y: $$3(-2) + 2y = 18 ightarrow -6 + 2y = 18 ightarrow 2y = 24 ightarrow y = 12$$ Thus, the coordinates of intersection are: $(-2, 12)$.