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To solve this problem, we need to define variables for Ronan's and Kelsey's numbers and create equations based on the provided conditions.

Let:
- \( r \) be Ronan's number.
- \( k \) be Kelsey's number.

### Step 1: Translate the conditions into equations
1. "2 less than Ronan's number is equal to 4 lots of Kelsey's number":
   \[
   r - 2 = 4k
   \]
2. "3 lots of Ronan's number added to Kelsey's number is 97":
   \[
   3r + k = 97
   \]

### Step 2: Solve the simultaneous equations
We have the system of equations:
\[
r - 2 = 4k \tag{Equation 1}
\]
\[
3r + k = 97 \tag{Equation 2}
\]

### Step 3: Solve for \( r \) and \( k \)
From Equation 1:
\[
r - 2 = 4k \implies r = 4k + 2
\]

Substitute \( r = 4k + 2 \) into Equation 2:
\[
3(4k + 2) + k = 97
\]
\[
12k + 6 + k = 97
\]
\[
13k + 6 = 97
\]
\[
13k = 91 \implies k = 7
\]

Now, find \( r \):
\[
r = 4k + 2 = 4(7) + 2 = 28 + 2 = 30
\]

### Solution
- Ronan's number: \( r = 30 \)
- Kelsey's number: \( k = 7 \)

Would you like more details on any step, or do you have any questions?

### Follow-Up Questions
1. How can we verify that these solutions satisfy the original conditions?
2. What method can be used to solve a system of equations with more than two variables?
3. How would the equations change if the problem mentioned "3 less than Ronan's number" instead?
4. Can you graph the equations \( r - 2 = 4k \) and \( 3r + k = 97 \) to find the solution visually?
5. How do the solutions change if Kelsey's number is increased by 2?

### Tip
When solving simultaneous equations, substitution is a great method if one of the variables is easily isolated.

Ronan and Kelsey each think of a number. 2 less than Ronan's number is equal to 4 lots of Kelsey's number. 3 lots of Ronan's number added to Kelsey's number is 97. Using the information above, write and solve two simultaneous equations to work out Ronan and Kelsey's numbers.

Learn how to solve a system of simultaneous equations where Ronan's and Kelsey's numbers are determined through algebraic equations. This example demonstrates the use of the substitution method to find the values of two variables, suitable for students in Grades 8-10.

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