Write an equation with a variable on both sides of the equal sign that has infinitely many solutions. Solve the equation and explain why it has an infinite number of solutions.
Okay so that is the question, I know it's already on Jiskha but I need it to be explained more. How would I know that a equation has infinite solutions.
That's the things I don't get, I don't know how to solve that. All I think I know is that the problem is basically 1=1.
x=x
What is the answer to this? Does it have a single answer? Or an infinite number of answers?
To write an equation with infinitely many solutions, we can choose a simple equation with the same variable on both sides of the equal sign. Let's consider the equation:
2x = 2x
In this equation, both sides are equal, so any value of x would make the equation true. Therefore, it has infinitely many solutions.
To understand why this equation has infinite solutions, we can take a closer look at the properties of equality. In this equation, we can perform the same operation on both sides and still maintain equality.
For example, if we divide both sides by 2:
(2x)/2 = (2x)/2
x = x
No matter what value we choose for x, it will always be equal to itself. So, every value of x would satisfy the equation, resulting in infinitely many solutions.
In general, an equation with the same variable on both sides will have an infinite number of solutions if the coefficients (numbers multiplied with the variable) on both sides are equal, or if the variable terms are canceled out through simplification.
To determine whether an equation has infinitely many solutions, we need to analyze the equation and see if it satisfies certain conditions.
An equation with infinitely many solutions usually occurs when both sides of the equation are equal regardless of the value of the variable. In other words, the equation is always true, regardless of the value of the variable.
Let's consider an example to illustrate this. Suppose we have the equation:
2x = 2x + 3
This equation has a variable (x) on both sides of the equal sign. To solve it, we can start by simplifying both sides:
2x - 2x = 2x + 3 - 2x
0 = 3
In this case, we ended up with 0 = 3, which is a false statement.
This means that there are no solutions for this equation. When both sides of the equation result in a false statement, it indicates that there are no common values for the variable that satisfy the equation. Therefore, this equation does not have infinitely many solutions.
Let's now consider an example that does have infinitely many solutions:
3x - 2 = x + 4
To solve this equation, we can start by simplifying both sides:
3x - x = 4 + 2
2x = 6
Now, to solve for x, we divide both sides of the equation by 2:
2x/2 = 6/2
x = 3
In this case, we obtained a value (x = 3) that satisfies the equation.
To see why this equation has infinitely many solutions, we can substitute any value for x into the equation and see that it holds true.
For example, if we substitute x = 5 into the equation:
3(5) - 2 = 5 + 4
13 = 9
As you can see, the equation does not hold true, which means that x = 5 is not a solution for the equation.
However, if we substitute x = 3, which is the value we obtained earlier:
3(3) - 2 = 3 + 4
7 = 7
Now, the equation holds true.
This demonstrates that for any real number we plug in for x, the equation will always hold true. Therefore, the equation has an infinite number of solutions.
In summary, an equation has infinitely many solutions when both sides of the equation are equal regardless of the value of the variable. This is confirmed by substituting different values for the variable and seeing whether the equation remains true.