Two lines are given by 2x+3y=12 and 4x−6y=18. Find their point of intersection. Are the lines parallel, perpendicular or neither?
Answer
Intersection at (21/4, 1/2); the lines are neither parallel nor perpendicular.
The correct answer is Intersection at (21/4, 1/2); the lines are neither parallel nor perpendicular. — Since m1≠m2 and m1⋅m2≠−1, the lines are neither parallel nor perpendicular.
Step-by-step solution
- Solve the system: multiply the first equation by 2 to get 4x+6y=24.
- Add the second equation 4x−6y=18 to the modified first: 8x=42 ⇒ x=21/4.
- Substitute x into the first equation: 2(21/4)+3y=12 ⇒ 21/2+3y=12.
- Solve for y: 3y=12−10.5=1.5 ⇒ y=1/2.
- Find slopes: for 2x+3y=12 → y=4−(2/3)x, slope m1=−2/3.
- For 4x−6y=18 → y=(2/3)x−3, slope m2=2/3.
- Since m1≠m2 and m1⋅m2≠−1, the lines are neither parallel nor perpendicular.
Common mistake: Assuming lines are parallel because coefficients look similar.
Concept tested: Intersection of two lines and classification
Published · How this solution was prepared and checked
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