• How do you know when an equation with one variable has infinitely many solutions?
• How do you know when an equation with one variable has no solution?
• How do you know when an equation with one variable has one solution?
• Give an example of each type of equation?
In order to determine how many solutions an equation with one variable has, you need to understand some key principles.
1. Infinitely Many Solutions: An equation with one variable has infinitely many solutions if any value you substitute for the variable satisfies the equation. In other words, no matter what value you choose, it will make the equation true.
2. No Solution: An equation with one variable has no solution if there is no value that can be substituted for the variable to make the equation true. This means that no matter which value you choose, the equation will always be false.
3. One Solution: An equation with one variable has one solution if there is only one specific value that can be substituted for the variable to make the equation true. This means that no other value will satisfy the equation.
Now, let's see some examples to illustrate each type:
1. Infinitely Many Solutions:
Consider the equation: 3x + 6 = 9.
If we solve this equation by subtracting 6 from both sides and dividing by 3, we get: x = 1.
Now, let's substitute any value for x, say 2: 3(2) + 6 = 12. This is still equal to 9. By substituting any other number, we will continue to get 9. Therefore, this equation has infinitely many solutions.
2. No Solution:
Consider the equation: 2x + 5 = 2x + 7.
By applying algebraic manipulations, we can simplify it to: 5 = 7, which is clearly false. No matter what number we substitute for x, the equation will never be true. Hence, this equation has no solution.
3. One Solution:
Consider the equation: 4x - 3 = 5.
By isolating x, we find that x = 2.5 when we solve for x.
No other value will make this equation true; if we substitute any other value for x, we will not obtain the left side equal to the right side. Thus, this equation has exactly one solution.
By understanding the concepts behind each type of equation, you can analyze equations with one variable and determine the number of solutions they possess.