It seems that the image file could not be accessed properly. Let me try again and read the math equations from the image.The image contains two separate linear equations, and you are asked to determine whether each equation has no solution, a single solution, or infinitely many solutions (i.e., "all real numbers are solutions").

1. First Equation:

$3(u + 1) + 7u = 4(u + 1) + 7$

Let's simplify both sides:

• Left-hand side: $3(u + 1) + 7u = 3u + 3 + 7u = 10u + 3$
• Right-hand side: $4(u + 1) + 7 = 4u + 4 + 7 = 4u + 11$

Now, equate both sides: $10u + 3 = 4u + 11$

Subtract $4u$ from both sides: $6u + 3 = 11$

Subtract 3 from both sides: $6u = 8$

Divide by 6: $u = \frac{8}{6} = \frac{4}{3}$

Thus, the solution to the first equation is: $u = \frac{4}{3}$

2. Second Equation:

$2(w + 2) - w = 3(w - 1) + 9$

Simplify both sides:

• Left-hand side: $2(w + 2) - w = 2w + 4 - w = w + 4$
• Right-hand side: $3(w - 1) + 9 = 3w - 3 + 9 = 3w + 6$

Equate both sides: $w + 4 = 3w + 6$

Subtract $w$ from both sides: $4 = 2w + 6$

Subtract 6 from both sides: $-2 = 2w$

Divide by 2: $w = -1$

Thus, the solution to the second equation is: $w = -1$

Summary:

1. For the first equation, the solution is $u = \frac{4}{3}$.
2. For the second equation, the solution is $w = -1$.

Would you like further details or explanations on these solutions?

Here are 5 related questions to deepen your understanding:

1. How can you check if a solution satisfies the original equation?
2. What is the geometric interpretation of finding no solution in a system of equations?
3. Can two linear equations ever have more than one distinct solution?
4. How do you handle equations involving fractions during solving?
5. Why is it important to simplify both sides of an equation before solving?

Tip: Always double-check your final solution by substituting it back into the original equation to verify it satisfies both sides.