To solve this problem, let's start by translating the given information into a system of equations. Sara and Rolando's ages are represented by [tex]\(S\)[/tex] and [tex]\(R\)[/tex] respectively.
1. We know that Sara is 33 years younger than Rolando.
- This relationship can be written as: [tex]\(R - S = 33\)[/tex]
2. We are also informed that the sum of their ages is 105.
- This gives us the equation: [tex]\(S + R = 105\)[/tex]
So, the system of equations that represents this situation is:
[tex]\[
\begin{array}{l}
S + R = 105 \\
R - S = 33
\end{array}
\][/tex]
Now that we have our system of equations, let's find the solution step-by-step:
### Step 1: Arrange the equations
[tex]\[
\begin{array}{l}
1. \quad S + R = 105 \\
2. \quad R - S = 33
\end{array}
\][/tex]
### Step 2: Add the two equations to eliminate [tex]\(S\)[/tex]
[tex]\[
(S + R) + (R - S) = 105 + 33
\][/tex]
### Step 3: Simplify the equation
[tex]\[
R + R = 138 \\
2R = 138
\][/tex]
### Step 4: Solve for [tex]\(R\)[/tex]
[tex]\[
R = \frac{138}{2} \\
R = 69
\][/tex]
### Step 5: Substitute [tex]\(R = 69\)[/tex] back into the first equation [tex]\(S + R = 105\)[/tex]
[tex]\[
S + 69 = 105
\][/tex]
### Step 6: Solve for [tex]\(S\)[/tex]
[tex]\[
S = 105 - 69 \\
S = 36
\][/tex]
Therefore, the solution to the system of equations is [tex]\(S = 36\)[/tex] and [tex]\(R = 69\)[/tex].
So, Sara is 36 years old and Rolando is 69 years old.
