https://file.aiquickdraw.com/mathbot/file/17265081793514quwhj3x.jpg

The equation in the image is:

\[
3(5x + 7) = 5x + (\text{missing value}).
\]

You are asked to determine the missing values that would make this equation have an infinite number of solutions.

### Step-by-Step Solution:

1. **Expand the left-hand side of the equation:**
   \[
   3(5x + 7) = 15x + 21.
   \]

2. **Rewrite the equation with the expanded form:**
   \[
   15x + 21 = 5x + (\text{missing value}).
   \]

For this equation to have an infinite number of solutions, both sides must be identical. Therefore, the terms involving \(x\) must be the same, and the constant terms must be the same.

3. **Set the coefficients of \(x\) equal:**
   \[
   15x = 5x.
   \]
   This implies the missing value on the right must include a term that adds \(10x\) to make both sides equal in terms of \(x\).

4. **Set the constant terms equal:**
   \[
   21 = \text{missing constant value}.
   \]
   This implies the constant value on the right must be 21.

### Conclusion:
The missing values that would make the equation have an infinite number of solutions are:
- The term next to \(5x\) should be \(10x\),
- The constant should be \(21\).

Thus, the completed equation would be:
\[
3(5x + 7) = 5x + 10x + 21.
\]

Would you like further explanation on any part of this? Here are some related questions:

1. What does it mean for an equation to have an infinite number of solutions?
2. How do you solve for equations with infinite solutions or no solutions?
3. What other conditions can make an equation have no solutions?
4. Could you generalize how to balance any linear equation to have infinite solutions?
5. What is the difference between solving linear equations with one solution, no solution, and infinite solutions?

**Tip:** For an equation to have infinite solutions, both sides must be identical, not just in form but also in numerical value for every term.

Determine the missing values that would make this equation have an infinite number of solutions: 3(5x + 7) = 5x + __ + __.

Learn how to solve the linear equation 3(5x + 7) = 5x + __ + __ to make it have infinite solutions. This problem explores concepts like linear equations, identity, and infinite solutions, suitable for Grades 7-9.

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