Three ways two lines can sit
Cross once, never meet, or be the same line.
A second line
Draw the line x + 2y − 4 = 0. Now draw a second line, any line you like.
How can the second line sit, compared with the first?
What this lesson covers
The idea
If the two lines intersect in a single point the pair has a unique solution (consistent); if they are parallel it has no solution (inconsistent); if they coincide it has infinitely many solutions (dependent, consistent).
A second line
Draw the line x + 2y − 4 = 0. Now draw a second line, any line you like.
How can the second line sit, compared with the first?
- It must cross the first one
- It can cross it, run alongside it, or lie on it
- It can only run alongside it
Move the orange line
Use + and − to change a₂, b₂ and c₂. Make all three pictures. Hint: changing only c₂ slides the line without turning it.
Three pictures
Two lines can be… intersecting: one common point, so a unique solution (consistent) parallel: no common point, so no solution (inconsistent) coincident: the same line, so infinitely many solutions (dependent, consistent)
These are the pairs from the dots. Pair 1 gave lines that intersect (at (4, 2)), pair 2 gave lines that coincide, and pair 3 gave lines that are parallel.
Two straight lines cannot meet in exactly two points. They meet once, never, or everywhere.
Notes
Intersecting: one solution (consistent). Parallel: no solution (inconsistent). Coincident: infinitely many solutions (dependent, consistent).
Check yourself
Two rails are x + 2y − 4 = 0 and 2x + 4y − 12 = 0. Will the rails cross each other?
2 pencils and 3 erasers cost ₹9, and 4 pencils and 6 erasers cost ₹18. The lines of 2x + 3y = 9 and 4x + 6y = 18 coincide. Can we find the price of one pencil?
Which picture goes with a pair that has exactly one solution?
The lines y = 2x − 2 and y = 4x − 4 have different steepness (2 and 4). In how many points do they meet?
Answer: 1
Different steepness means not parallel and not the same line, so they cross once: at (1, 0), as in Champa’s problem.
- No: the lines are parallel, so there is no common solution — correct. Yes! Divide the second by 2: x + 2y = 6 can never equal x + 2y = 4.
- Yes: at one point. One crossing point would be one common solution. There is none.
- Yes: they lie on top of each other. Then they would share every point. These two run side by side.
- No: many pairs (x, y) fit, so the price is not fixed — correct. Yes! Coincident lines mean infinitely many solutions.
- Yes: where the lines cross. They do not cross at a single point. They lie on top of each other.
- No: the lines are parallel. Parallel lines never meet. These lines meet everywhere.
- Intersecting lines — correct. Yes! One common point means one solution.
- Parallel lines. Parallel lines share no point: no solution.
- Coincident lines. Coincident lines share every point: infinitely many solutions.