https://file.aiquickdraw.com/mathbot/file/17266927622525hjbjneb.jpg

To solve the problem, we need to find the number in the box that makes the equation have no solutions.

The equation is:
\[
\text{[Box]} + x + 13 = -5x - 2 + 7x
\]

### Step-by-step solution:

1. **Simplify both sides:**

   - On the right-hand side, combine like terms:
   \[
   -5x + 7x = 2x
   \]
   So the equation becomes:
   \[
   \text{[Box]} + x + 13 = 2x - 2
   \]

2. **Move all terms involving \( x \) to one side:**
   Subtract \( x \) from both sides:
   \[
   \text{[Box]} + 13 = x - 2
   \]

3. **Set up for no solutions:**
   For this equation to have no solutions, the constant terms must be unequal while the coefficients of \( x \) must cancel each other. This happens when we get a contradiction, such as a false statement like \( 5 = 2 \).

   To achieve this, the coefficient of \( x \) on both sides must be the same. So, for the equation to have no solution, the constant on the left side and right side should differ.

4. **Find the missing number:**
   For there to be no solution, the number in the box must make the constant terms unequal. We see that the right-hand side constant is \( -2 \), and for no solution, the constant on the left must differ. So we set:
   \[
   \text{[Box]} + 13 \neq -2
   \]

   Solving for the missing number:
   \[
   \text{[Box]} \neq -2 - 13
   \]
   \[
   \text{[Box]} \neq -15
   \]

So, the missing number is **not** \(-15\) in order for the equation to have no solutions.

---

Would you like any further details or clarification on this? 

Here are 5 related questions for deeper understanding:

1. What does it mean for an equation to have no solutions?
2. How can we distinguish between an equation with no solutions and one with infinite solutions?
3. What happens if the number in the box is \(-15\)?
4. Can this equation have exactly one solution? If yes, how?
5. How does the coefficient of \( x \) affect the number of solutions in an equation?

**Tip:** Always check for contradictions when simplifying equations to determine if they have no solutions!

Find the missing number so that the equation has no solutions: □ + x + 13 = -5x - 2 + 7x.

Learn how to find the missing number in the equation □ + x + 13 = -5x - 2 + 7x so that it has no solutions. This algebra problem explains key concepts such as combining like terms and checking for no solutions. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation