The lines x+y = 10 and -3x+y = 2 have a unique solution. What is the distance (in units) between the unique solution and the point of intersection of the line x+y = 10 and the x-axis?
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Correct answer: A
Explanation
The correct answer is option 1 i.e. .
x + y = 10
-3x + y = 2
Subtract the second equation from the first: (x + y) - (-3x + y) = 10 - 2
⇒ 4x = 8
⇒ x = 2
Substitute x = 2 into the first equation:
⇒ 2 + y = 10
⇒ y = 8
Solve the first equation for y: y = 10 - x
Substitute this expression for y in the second equation:
⇒ -3x + (10 - x) = 2
⇒ -4x = -8
⇒ x = 2
Substitute x = 2 into the first equation to find y: y = 10 - 2 = 8
The unique solution is the point (2, 8).
Find the point of intersection of the line x + y = 10 and the x-axis:
On the x-axis, y = 0.
Substitute y = 0 into the equation:
⇒ x + y = 10
⇒ x + 0 = 10
⇒ x = 10
The point of intersection is (10, 0).
Calculate the distance between the two points:
Use the distance formula: Distance = √[(x2 - x1)² + (y2 - y1)²]
⇒ Distance = √[(10 - 2)² + (0 - 8)²]
⇒ Distance = √[8² + (-8)²]
⇒ Distance = √(64 + 64)
⇒ Distance = √128
⇒ Distance = 8√2
Therefore, the distance between the unique solution (2, 8) and the point of intersection of the line x + y = 10 and the x-axis (10, 0) is 8√2 units.