Understanding Equations with No Solution
In mathematics, an equation with no solution occurs when the variables cancel each other out, leading to a false statement. Let’s explore the equation derived from your topic step-by-step.
Given Expression
We have the expression:
4x + 2 = 6 + 2x + 9 + 3x + 6 + 6 + 2x + 8
Simplifying the Right Side
First, let’s simplify the right side:
- Combine like terms on the right side:
- 6 + 9 + 6 + 6 + 8 = 35
- 2x + 3x + 2x = 7x
Thus, the equation becomes:
4x + 2 = 7x + 35
Rearranging the Equation
Next, we can move all the variables to one side and constants to the other:
- Subtract 7x from both sides:
- This simplifies to:
4x – 7x + 2 = 35
-3x + 2 = 35
Isolating x
Next, we’ll isolate x by subtracting 2 from both sides:
-3x = 33
Now, divide by -3:
x = -11
Conclusion: No Solution Scenario
If we transform the original equation and you arrive at a statement that is clearly false, such as:
0 = 5
Then, it indicates that there is no solution because you cannot have any value for x that satisfies this equation.
Example of No Solution
As an example, if we were to manipulate equations and arrive at:
4x + 3 = 4x + 7
Subtracting 4x from both sides leads to:
3 = 7
This false statement shows that the original equation has no solution.
Final Thoughts
In summary, not all equations yield solutions, especially when they boil down to contradictory statements. In your case, we manipulated the equation to show the principles behind discovering no solutions in algebra.