Math 1 · Unit 6 · Lesson 4 · part 3 of 5
What a strange final line means
When you solve by substitution or elimination, the algebra tells the same story, and it is worth knowing...
What a strange final line means
When you solve by substitution or elimination, the algebra tells the same story, and it is worth knowing what a strange final line means.
One solution: you get x = a number
y = 2x − 1 and y = −x + 5. Substitute: 2x − 1 = −x + 5, so 3x = 6, so x = 2, then y = 3. A normal ending: the variable has one value.
No solution: a false line like 0 = 5
y = 2x + 1 and y = 2x − 3. Substitute: 2x + 1 = 2x − 3. Add −2x to both sides: 1 = −3. The x disappeared and what is left is false. No x can make it true, so there is no solution. The lines are parallel.
Infinitely many: a true line like 0 = 0
y = 2x + 1 and 2y = 4x + 2. Substitute: 2(2x + 1) = 4x + 2, so 4x + 2 = 4x + 2, so 0 = 0. The x disappeared and what is left is always true. Every x works, so every point on the line is a solution. It is the same line twice.