To model the given situation with [tex]\( y \)[/tex] representing the larger number and [tex]\( x \)[/tex] representing the smaller number, we need to set up equations based on the statements provided:
1. Statement 1: A larger number is double the sum of 3 and a smaller number.
Let's represent this mathematically. The phrase "double the sum of 3 and a smaller number" translates to:
[tex]\[
y = 2(x + 3)
\][/tex]
This equation expresses that [tex]\( y \)[/tex] (the larger number) is twice the sum of [tex]\( x \)[/tex] (the smaller number) and 3.
2. Statement 2: The larger number is 2 less than 3 times the smaller number.
Representing this mathematically:
[tex]\[
y = 3x - 2
\][/tex]
This equation expresses that [tex]\( y \)[/tex] is 2 less than 3 times [tex]\( x \)[/tex].
Now, let's examine other possible forms of these equations for completeness.
3. Rearranging equation [tex]\( y = 3x - 2 \)[/tex] into a standard form, we get:
[tex]\[
3x - y = 2
\][/tex]
4. Equation [tex]\( y = 3x - 2 \)[/tex] can also be written as:
[tex]\[
y \neq 2 - 3x
\][/tex]
This form does not represent the given situation correctly because it does not align with the original conditions provided. Hence, it is incorrect.
5. The equation [tex]\( 3x - y = -2 \)[/tex] would imply:
[tex]\[
y = 3x + 2
\][/tex]
This would mean the larger number is 2 more than 3 times the smaller number, which contradicts the given statement. Hence, it is also incorrect.
Hence, the equations that correctly model the situation are:
- [tex]\( y = 2(x + 3) \)[/tex]
- [tex]\( y = 3x - 2 \)[/tex]
- [tex]\( 3x - y = 2 \)[/tex]
These equations are the ones that correctly describe the relationship between the larger number [tex]\( y \)[/tex] and the smaller number [tex]\( x \)[/tex] according to the given conditions. Therefore, the correct options are:
- [tex]\( y = 3x - 2 \)[/tex]
- [tex]\( 3x - y = 2 \)[/tex]
- [tex]\( y = 2(x + 3) \)[/tex]
