When the letter vanishes

The idea

If substitution or elimination leaves a statement with no variable, a true statement means the pair has infinitely many solutions and a false statement means the pair is inconsistent (no solution).

Where did y go?

Take x + 2y = 4 and 2x + 4y = 8. Write x = 4 − 2y and put it into the second equation: 2(4 − 2y) + 4y = 8.

Open the brackets and collect. What is left?

• y = a number
• A statement with no letter in it
• Nothing at all

Substitute and look

All three pairs start with x + 2y = 4. Put x into equation (2), read the last line, and say how many solutions there are. Then check on the picture.

A statement with no letter

If substitution (or elimination) leaves a statement with no variable: a true one (like 8 = 8) means infinitely many solutions a false one (like 8 = 12) means the pair is inconsistent: no solution

Why? A true statement holds whatever y is, so every y works. A false statement holds for no y, so nothing works.

The book meets the same thing in Example 6 (18 = 18, the pencils and erasers) and Example 7 (−4 = 0, the rails).

Notes

A statement with no variable: true means infinitely many solutions; false means no solution (inconsistent).

Check yourself

Solving a pair leaves 5 = 5. What does that tell us?

Elimination on 2x + 3y = 8 and 4x + 6y = 7 (Example 9) leaves 0 = 9. What does that tell us?

A pair leaves the statement 3 = 7 after substitution. How many solutions does it have?

Answer: 0

3 = 7 is false, so no values of x and y satisfy both equations: 0 solutions.

In the pencil problem (2x + 3y = 9 and 4x + 6y = 18) substitution leaves 18 = 18. Why can we not find one price for a pencil?

• The pair has infinitely many solutions — correct. Yes! A true statement with no letter means the two equations say the same thing.
• The pair has no solution. A false statement would mean no solution. 5 = 5 is true.
• x = 5 and y = 5. There is no x or y in 5 = 5, so it gives no values.
• No solution: 0 = 9 is false — correct. Yes! The pair is inconsistent.
• Infinitely many solutions. That needs a true statement, like 0 = 0.
• x = 0. 0 = 9 has no letter, so it gives no value of x.
• Both equations say the same thing, so many prices fit — correct. Yes! Infinitely many solutions.
• One of the equations is wrong. Neither is wrong. They give the same information.
• The price is 18. 18 = 18 has no letter in it. It is not a price.