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If there are less distinct equations than there are variables then there will be an infinite number of solutions.
For example, you may have 3 equations with 3 unknowns, but if one of those equations is a multiple of another there there are only 2 distinct equations:
- 2x + 3y + 5z = 1
- x + y - 2z = 10
- 4x + 6y + 10z = 2
If any two equations are parallel then there is no solution; if equation (3) above was 2x + 3y + 5z = 2, then there are no solutions - subtracting equation 1 from (the new) equation 3 would result in 0 = 1 which is nonsense.
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How do you know when an equations has infinitly many solutions?
An equation must have 1, 0, or infinitely many solutions. So if you find 1 and there is another, you have know it has infinitely many. For example. 0x+2=2 I solve this and the equations become 0x=0 Now, 1 is a solutions, but so is 2. I now know there are infinitely many. How about 0x+2=3. No solution and 2x+2=4, has one solution. I put those two here so you might try other numbers and see that they have no solutions and one solution. A special type of equation known as an identity is an equation that holds for all numbers. This means it has infinitely many solutions.
How can you tell when an equation in one variable has infinitely many solutions or no solutions?
There is no simple method. The answer depends partly on the variable's domain. For example, 2x = 3 has no solution is x must be an integer, or y^2 = -9 has no solution if y must be a real number but if it can be a complex number, it has 2 solutions.
How do you know when an Equation has an infinity solutions without solving the equation?
You can't really know that in all cases. But with some practice in working with equations, you'll start to notice certain patterns. For example, you'll know that certain functions are periodic, and that an equation such as: sin(x) = 0 have infinitely many solutions, due to the periodicity of the function. This one is easy; we can make some small changes: sin(2x + 3) = 0.5 Here it isn't as easy to guess the exact solutions of the equation, but due to our knowledge of the periodicity of the sine function, we can assume that it has infinitely many solutions. Another example: a single equation with two or more variables normally has infinitely many solutions, for example: y = 3x + 2
How do you find the linear system if you know its solutions?
You cannot since there are infinitely many sets of lines that can pass through any single point - the solution.
Why is it important to know various techniques for solving systems of equations and inequalities?
It is important to know several techniques for solving equations and inequalities because one may work better than another in a particular situation.
