Exploring the Nature of Solutions
Solve the following equations. What do you notice/wonder about the solutions?
1. \(2x – 3+ 1 = 2(x – 1)\)
2. \(5(x – 1) = 3(x + 2)\)
3. \(4(x – 2) + 2 = x + 3(x – 2)\)
Solutions: One solution, no solution, and infinitely many solutions respectively.
Notes: A linear equation can have one solution, no solution, or an infinite number of solutions.
- An equation of the form \(x+ a = x + a\) has infinite solutions since you can use any value for \(x\) and this statement will always be true.
- An equation of the form \(x + a = x – a\) has no solutions since there is no value for \(x\) that will make this statement true.
- An equation of the form \(x + a = b\) has one solution since there is only one value for \(x\) that will make this statement true.