Let's solve the equation \((m+n)x = 5x - 3 + x + 4\).

1. Simplify the right-hand side of the equation:
   \[
   5x - 3 + x + 4 = 6x + 1
   \]

2. Rewrite the equation:
   \[
   (m+n)x = 6x + 1
   \]

3. For the equation to hold true for all values of \(x\), the coefficients of \(x\) on both sides must be equal, and the constant terms must also be equal. Thus, we have:
   \[
   m+n = 6
   \]
   and there should be no constant term on the left side to match the constant term on the right side:
   \[
   0 = 1
   \]

However, this indicates a contradiction because the constant term \(1\) on the right side has no counterpart on the left side. This implies that the given equation \((m+n)x = 6x + 1\) does not have a valid solution due to the constant term mismatch.

Would you like further details or have any other questions?

Here are some related questions:
1. How do you handle inconsistent equations in algebra?
2. What are the steps to verify the solution of an equation?
3. Can you solve a system of linear equations with variables on both sides?
4. How do you interpret the solutions to equations with no valid solutions?
5. What are the methods to isolate variables in linear equations?
6. Can you provide examples of consistent equations with solutions?
7. What is the significance of the equality of coefficients in polynomial equations?
8. How do you solve equations involving polynomials of higher degrees?

**Tip:** Always compare both the coefficients and constant terms on both sides of an equation to check for consistency and possible solutions.

Learn how to solve the equation (m+n)(x) = 5x - 3 + x + 4 with a detailed step-by-step solution. This problem involves solving a linear equation with variables and constants on both sides. Suitable for students in Grades 7-9.

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