Consistent and inconsistent pairs

Three ways two lines can sit

Two straight lines on a page can be in one of three positions.

• The lines cross. They share exactly one point, so the pair has exactly one solution. The pair is consistent.
• The lines are parallel. They share no point, so the pair has no solution. The pair is inconsistent.
• The lines coincide. They share every point, so the pair has infinitely many solutions. The pair is consistent, and its equations are dependent.

A pair is consistent when it has at least one solution. Two different lines can meet only once, so a consistent pair has either one solution or infinitely many. It can never have exactly two.

What the three cases mean in words

A shopkeeper tells Asha: "2 pens and 3 notebooks cost ₹30." Later she says: "4 pens and 6 notebooks cost ₹60."

• The second sentence is the first one doubled. It adds no new information. Many prices for a pen and a notebook fit both sentences. These equations are dependent.
• If the shopkeeper had said "4 pens and 6 notebooks cost ₹50", she would contradict herself, because twice ₹30 is ₹60. No prices fit both sentences. The pair is inconsistent.
• If she had said "3 pens and 2 notebooks cost ₹25", the two sentences would tell Asha something new. Exactly one pair of prices fits both.

Three pairs, three drawings

1. Crossing lines. $x + y = 5$ and $2x - y = 1$. You cannot turn the left side of one into the left side of the other by multiplying. The lines cross at $(2, 3)$: one solution.

2. Parallel lines. $x + 2y = 4$ and $2x + 4y = 12$. Divide the second by $2$ and it becomes $x + 2y = 6$. Now one left side would have to be both $4$ and $6$. No point can do that, so there is no solution.

3. Coincident lines. $2x + 3y = 9$ and $4x + 6y = 18$. The second is the first multiplied by $2$, so every point of one line is on the other: infinitely many solutions.

The idea behind all three: multiply one equation to match the left sides. If the right sides then differ, the lines are parallel. If the right sides match, the lines coincide. If the left sides cannot be matched, the lines cross.

Practise with new numbers

Try it in Python

This program draws three pairs side by side and says how many solutions each pair has.

Common mistakes

• Saying "no solution" for coincident lines. They have infinitely many solutions, not none.
• Thinking two equations that look different must give different lines. Multiply one by a number: $2x + 3y = 9$ and $4x + 6y = 18$ are the same line.
• Calling a pair with no solution "dependent". Dependent means the lines coincide.
• Judging parallel lines by eye. Two lines can look parallel and still meet far away. The next lesson gives an exact test.

Remember these

• Crossing lines: one solution, consistent.
• Parallel lines: no solution, inconsistent.
• Coincident lines: infinitely many solutions, consistent and dependent.
• A pair can never have exactly two solutions.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A pair has parallel lines. How many solutions does it have, and what is the pair called?

2. Do $x + y = 4$ and $3x + 3y = 12$ give the same line?

3. Is the pair $x + y = 4$ and $x + y = 7$ consistent?

4. Can a consistent pair have exactly two solutions?

5. What word describes the equations of a pair whose lines coincide?