o How do you know when an equation has an infinite number of solutions?
o How do you know when an equation has no solution?
If after you perform all proper steps and simplifications, the variables drop out and you end up with a TRUE statement, then there is an infinite number of solutions.
If after you perform all proper steps and simplifications, the variables drop out and you end up with a FALSE statement, then there is no solution.
e.g.
(x+1)^2 = x^2 + 2x + 1
x^2 + 2x + 1 = x^2 + 2x + 1
1=1 -----true, thus infinite numer of solutions
e.g.
3(2x + 5) = 6x+1
6x + 15 = 6x+1
15 = 1 ---- false, therefore no solution.
To determine if an equation has an infinite number of solutions or no solution, you need to analyze the equation and compare the coefficients and constant terms. Let's break it down for each case:
1. Infinite number of solutions:
An equation has an infinite number of solutions when it represents the same line or when all the variables cancel out. To identify this, you need to check if the coefficients of all variables are proportional on both sides of the equation. In other words, if the equation can be reduced to a statement like 3x + 2y = 6, where the equation can be represented as a single line.
Example:
Consider the equation 3x + 6y = 12. By dividing both sides of the equation by 3, you can reduce it to x + 2y = 4. In this case, since you can choose infinite values for x and y that satisfy the equation, it has an infinite number of solutions.
2. No solution:
An equation has no solution when it represents parallel lines or contradictory statements. To identify this, you need to check if the coefficients and constant terms are such that the equation becomes false or contradictory.
Example:
Consider the equation 2x + 3y = 10 and 2x + 3y = 5. Here, both equations have the same coefficients and different constant terms. Since no values of x and y satisfy both equations simultaneously, there is no solution.
In summary, by analyzing the coefficients and constant terms of an equation, you can determine whether it has an infinite number of solutions or no solution.