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Rachel is 3 years older than Sarah. Their ages add to 67. Which of the following systems correctly models this situation?

[tex]\[
\begin{array}{c}
\left\{
\begin{array}{l}
r - s = 3 \\
r + s = 67
\end{array}
\right. \\
\left\{
\begin{array}{l}
r - s = 67 \\
r + s = 3
\end{array}
\right. \\
\left\{
\begin{array}{l}
s + r = 3 \\
r - s = 67
\end{array}
\right. \\
\left\{
\begin{array}{l}
s - r = 3 \\
r + s = 67
\end{array}
\right.
\end{array}
\][/tex]

Sagot :

GinnyAnsweridn
Let's analyze the situation step by step to determine which system of equations correctly represents the facts given.

1. Chel is 3 years older than Sarah:
If we let [tex]\( r \)[/tex] represent Chel's age and [tex]\( s \)[/tex] represent Sarah's age, this relationship can be written as:
[tex]\[ r - s = 3 \][/tex]

2. The sum of their ages is 67:
This relationship can be written as:
[tex]\[ r + s = 67 \][/tex]

Now, let’s compare these equations to each of the given systems:

### System 1:
[tex]\[ \left\{\begin{array}{l} r - s = 3 \\ r + s = 67 \end{array}\right. \][/tex]
This set of equations correctly captures the relationships:
- [tex]\( r - s = 3 \)[/tex] indicates Chel is 3 years older than Sarah.
- [tex]\( r + s = 67 \)[/tex] indicates the sum of their ages is 67.

### System 2:
[tex]\[ \left\{\begin{array}{l} r - s = 67 \\ r + s = 3 \end{array}\right. \][/tex]
This set of equations is not correct because:
- [tex]\( r - s = 67 \)[/tex] incorrectly states that one is 67 years older than the other, which does not match the given relationship.
- [tex]\( r + s = 3 \)[/tex] incorrectly states that the sum of their ages is 3.

### System 3:
[tex]\[ \left\{\begin{array}{l} s + r = 3 \\ r - s = 67 \end{array}\right. \][/tex]
This set of equations is also incorrect because:
- [tex]\( s + r = 3 \)[/tex] incorrectly states that the sum of their ages is 3.
- [tex]\( r - s = 67 \)[/tex] incorrectly states that one is 67 years older than the other.

### System 4:
[tex]\[ \left\{\begin{array}{l} s - r = 3 \\ r + s = 67 \end{array}\right. \][/tex]
This set of equations is not correct because:
- [tex]\( s - r = 3 \)[/tex] incorrectly states that Sarah is 3 years older than Chel.
- Although [tex]\( r + s = 67 \)[/tex] is correct, the first equation must also be correct for the system to represent the original problem accurately.

From the analysis, the system that correctly models the given situation is:

[tex]\[ \left\{\begin{array}{l} r - s = 3 \\ r + s = 67 \end{array}\right. \][/tex]

Thus, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
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