Since I am new to the algebra and do not fully understand it, I am not sure how to answer the following question. Hope you can help. thanks
how do you know when an equation has infinitely many solutions
If after performing all valid steps, your variable terms all cancel and you end up with something like
0=0, or 8=8, that is, a true statement,
then you have an infinite number of solutions.
if after all variable terms drop out, you are left with a false statement, there is no solution.
e.g.
2x + 6 = 2(x+3+
2x +6 = 2x + 6+0=0 .....true ---> infinite number of solutions.
e.g
2x+6 = 2(x+5)
2x+6 = 2x+10
0 = 4 .... false ---> no solution
clearly a typo, should be
2x + 6 = 2(x+3)
2x +6 = 2x + 6 .....true ---> infinite number of solutions.
To determine if an equation has infinitely many solutions, you need to look at the equation itself and its properties. Here are a few ways to identify if an equation has infinitely many solutions:
1. Linear dependence: If all the variables in the equation disappear when you simplify it, you have an equation that is always true, meaning it has infinitely many solutions. For example, if you have the equation 2x + 3y = 2x + 3y, you can simplify it to 0 = 0, which is true no matter what values x and y take.
2. Equation of identity: If you have an equation where both sides are equal, regardless of the values of the variables, then it has infinitely many solutions. For example, the equation 3x + 5 = 3x + 5 is true for any value of x since both sides are always equal.
3. Overlapping lines: In a system of linear equations (multiple equations with the same variables), if the lines represented by the equations overlap or coincide, then the system has infinitely many solutions. This occurs when the equations are essentially the same or are scalar multiples of each other.
4. Simplification to the same expression: If you can simplify the equation to have the same expression on both sides, regardless of the values of the variables, then it has infinitely many solutions. For example, if you have the equation (x + 2)^2 = x^2 + 4x + 4, you can simplify it to x^2 + 4x + 4 = x^2 + 4x + 4. Both sides are the same expression, so the equation has infinitely many solutions.
Remember, these are just a few examples of scenarios where an equation has infinitely many solutions. By examining the properties and behavior of the equation, you can determine if it has a single solution, no solution, or an infinite number of solutions.