how do you know when an equation has infinate many solutions and how do you know if an equation has no solutions?
The second part of this question I get. If you work an equation and come up with 5=3 this is not a solution. Is that part right? I guess I do not understand the question.
For your second question, yes.
Let me see if I get the first part. If you have an equation and the solution is correct, then how does this make it infinate? Does this mean any integer you substitute for the variable will work? Can you explain this to me?
for the first part,, sorry, i am not sure about this,, if you only have ONE given equation for instance,, sin(x)=1 ==> the solution for this is x = pi*2+k*2pi, for k greater than or equal to zero (note: pi/2 is in radians which is also equal to 90 degrees), therefore sin x = 1 has infinitely many solutions,,
another example if only one equation is given is 1/x=0 ,,here no real value of x will satisfy this, thus it has no solution.
if you have TWO given LINEAR equations, check if they are parallel, intersecting, or coinciding,,
*if parallel==> no solution since they will never intersect
[for an equation, Ax+By=C, two equations are parallel if their respective A and B are equal
ex: 2x-5y=3 and 2x-5y=-8]
*if intersecting==> one unique solution, since they will intersect at one point
*if coinciding ==> infinitely many solutions since the two given equations are equal
[for an equation, Ax+By=C, two equations are coinciding if their respective A and B are the multiple of the other
ex: 3x-5y=4 and 6x-10y=8
so there,, i hope i was able to help you.. =)
To determine if an equation has infinite solutions, you need to analyze its structure and coefficients. An equation typically has infinite solutions if the equation is dependent on one or more variables, which means that removing any variable(s) from the equation still results in a true statement.
One way to recognize an equation with infinite solutions is when all terms on one side of the equation cancel out when simplified. This can lead to an equation such as 0 = 0. For example, in the equation 2x + 4 = 2(x + 2), you can simplify it to 2x + 4 = 2x + 4. Here, every x-value will satisfy the equation, so there are infinite solutions.
On the other hand, an equation has no solution if the equation is inconsistent. Inconsistent equations do not have any common solution for the variables involved. This occurs when simplifying the equation further results in a false statement. A common example is when you have an equation like 3x + 5 = 3x + 7. By subtracting 3x from both sides, you get 5 = 7, which is not true. Since there is no number that can satisfy this equation, there are no solutions.
You correctly noted that an equation such as 5 = 3 is not a solution. When solving an equation, you typically aim to find a value or values that make the equation true. If you end up with a statement like 5 = 3, which is false, it means that there are no solutions for that equation.
In summary, an equation has infinite solutions if it is dependent and simplifies to a true statement like 0 = 0. An equation has no solutions if it is inconsistent and simplifies to a false statement like 5 = 3.