Check whether the given pair of lines are Intersecting, Parallel or Coincident x-3y-7=0 3x-9y 16=0?

Question

Answer

X - 3y - 7 = 0 –> Equation(1).
3x - 9y - 16 = 0 –> Equation (2).

Multiply Equation (1) by the number 3.
3x - 9y - 14 = 0.

3x - 9y - 14 = 0 ==> a1x + b1y + c1 = 0.
3x - 9y - 16 = 0 ==> a2x + b2y + c2 = 0.

Above Equations are in the form of a1x + b1y + c1 = 0, a2x + b2y + c2 = 0.

a1 = 3, b1 = -9, c1 = -14.
a2 = 3, b2 = -9, c2 = -16.

a1/a2 = 3/3 = 1.
b1/b2 = -9/-9 = 1.
c1/c2 = -14/-16 = 7/8.

Therefore, a1/a2 = b1/b2 ≠ c1/c2.

The above pair of lines are parallel.

Answer

X - 3y - 7 = 0 => 3y = x - 7

m1 = 1/3

3x - 9y - 16 = 0 => 9y = 3x - 16

m2 = 1/3

Since m1 = m2 the given lines are parallel

(The graph given below is not to scale and only for sake of understanding)

Answer

Intersecting, Parallel or Coincident Lines

To determine whether the given pair of lines are intersecting, parallel, or coincident, we need to compare their slopes and y-intercepts. The general form of a linear equation is y = mx + c, where m represents the slope and c represents the y-intercept.

Given Equations:
Line 1: x - 3y - 7 = 0
Line 2: 3x - 9y - 16 = 0

Finding the Slopes:
To find the slopes of the lines, we need to rewrite the equations in slope-intercept form (y = mx + c).

Line 1:
x - 3y - 7 = 0
- 3y = -x + 7 (subtracting x from both sides)
y = (1/3)x - 7/3 (dividing by -3)

Line 2:
3x - 9y - 16 = 0
-9y = -3x + 16 (subtracting 3x from both sides)
y = (1/3)x - 16/9 (dividing by -9)

Comparing the Slopes:
The slopes of the given lines are both (1/3). If the slopes of two lines are equal, the lines are either parallel or coincident. To differentiate between these two possibilities, we need to compare the y-intercepts.

Finding the Y-intercepts:
Line 1:
y = (1/3)x - 7/3
The y-intercept is -7/3.

Line 2:
y = (1/3)x - 16/9
The y-intercept is -16/9.

Comparing the Y-intercepts:
The y-intercepts of the given lines (-7/3 and -16/9) are not equal. Therefore, the lines are not coincident.

Conclusion:
Since the slopes of the lines are equal but the y-intercepts are different, the given pair of lines are parallel to each other.

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