Let's denote the three numbers as $x$ (the first number), $y$ (the second number), and $z$ (the third number).

We are given the following information:

1. The sum of the three numbers is 116: $x + y + z = 116$

2. The first number is 9 more than the second: $x = y + 9$

3. The third number is 3 times the first: $z = 3x$

Step 1: Substitute the second and third equations into the first equation.

Substitute $x = y + 9$ and $z = 3x$ into the sum equation $x + y + z = 116$:

$(y + 9) + y + 3(y + 9) = 116$

Step 2: Simplify the equation.

Distribute and combine like terms:

$y + 9 + y + 3y + 27 = 116$ $5y + 36 = 116$

Step 3: Solve for $y$.

Subtract 36 from both sides:

$5y = 80$

Divide by 5:

$y = 16$

Step 4: Find $x$ and $z$.

Now that we know $y = 16$, we can find $x$ and $z$:

$x = y + 9 = 16 + 9 = 25$ $z = 3x = 3 \times 25 = 75$

Final Answer:

The three numbers are:

• First number ($x$): 25
• Second number ($y$): 16
• Third number ($z$): 75

Would you like more details or have any other questions?

Here are 8 related questions you might find helpful:

1. How can we solve a system of linear equations with more than two variables?
2. What are the methods to solve linear equations, and when should each be used?
3. How can substitution be applied to solve word problems involving algebra?
4. What are some examples of real-life situations where such systems of equations are used?
5. How do we check if the solution to a system of equations is correct?
6. What are the differences between dependent, independent, and inconsistent systems of equations?
7. Can we solve this problem using a matrix approach? How?
8. How do word problems help in understanding algebra better?

Tip: When solving word problems, always define the variables clearly and translate the problem into mathematical equations step by step.