How do you know when an equation has infinitely many solutions?
How do you know when an equation has no solution?
If the variable drops out, and you end up with a true statement, then there is an infinite number so solutions,
if the end statement is false, there is not solution
e.g.
2x - 4 = 2(x-2)
2x-4 = 2x-4
0 = 0 ..... infinite number of solutions
2x-4 = 2(x-5)
2x-4 = 2x-10
-4 = -10 ... false!, no solution
Thank you
X^2+8x+10=0
To determine whether an equation has infinitely many solutions or no solution, you analyze the relationship between the coefficients and variables in the equation. Here's how:
1. Infinitely Many Solutions:
- When solving an equation, if you find that every term cancels out during simplification on both sides of the equation, you have an identity, such as "0 = 0" or "3x - 2x = x-x." This means that all values of the variable would make the equation true. In this case, the equation has infinitely many solutions.
- Another way to identify infinitely many solutions is when the equation includes variables that have been eliminated, resulting in a statement like "0 = 0" or a true statement like "2x = 2x."
2. No Solution:
- If, during the process of solving the equation, you find contradictory terms that cancel each other out, resulting in an inconsistent statement, you have "0 = a nonzero constant" or "0 = 2," for example. This implies that there is no value of the variable that can satisfy the equation, meaning it has no solution.
- Similarly, if, while solving the equation, you obtain a false statement like "3x = 2x + 1" or "8 = 9," this means that no value of the variable will make the equation true, resulting in no solution.
Remember, these methods can be applied to linear equations with one variable as well as linear systems with multiple variables. By examining the relationship between the terms and constants in the equation, you can determine whether it has infinitely many solutions or no solution.