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Create a linear system to model this situation:

A length of outdoor lights is formed from strings that are [tex][tex]$6 \, \text{ft}$[/tex][/tex] long and [tex][tex]$12 \, \text{ft}$[/tex][/tex] long. Thirteen strings of lights are [tex][tex]$126 \, \text{ft}$[/tex][/tex] long.

a.
[tex]\[ x + y = 13 \][/tex]
[tex]\[ 6x + 12y = 13 \][/tex]
[tex]\[ 6x + 12y = 126 \][/tex]

b.
[tex]\[ x + y = 13 \][/tex]
[tex]\[ x + y = 126 \][/tex]
[tex]\[ x + 2y = 126 \][/tex]

c.
[tex]\[ x + y = 13 \][/tex]
[tex]\[ 6x + 12y = 126 \][/tex]

d.
[tex]\[ x + y = 13 \][/tex]
[tex]\[ 6x + 12y = 126 \times 13 \][/tex]

Sagot :

GinnyAnswer
To solve this problem, we need to model it using a system of linear equations. Let's define our variables:

- Let [tex]\( x \)[/tex] be the number of 6-foot strings.
- Let [tex]\( y \)[/tex] be the number of 12-foot strings.

We know two things:
1. The total number of strings is 13.
2. The total length of the strings is 126 feet.

Let's write these conditions as equations:

1. The number of strings:
[tex]\[ x + y = 13 \][/tex]

2. The total length of the strings:
[tex]\[ 6x + 12y = 126 \][/tex]

Therefore, the correct system of equations to model the situation is:
[tex]\[ x + y = 13 \][/tex]
[tex]\[ 6x + 12y = 126 \][/tex]

Let's find the correct option among the given ones.

a. [tex]\( x + y = 13 \)[/tex]
- This option only provides one of the required equations, so it is incomplete.

b. [tex]\( x + y = 13 \)[/tex]
- [tex]\( x + y = 126 \)[/tex]
- [tex]\( x + 2y = 126 \)[/tex]
- This option is incorrect because the second and third equations do not fit the given information, and the first equation is repeated.

c. [tex]\( 6x + 12y = 13 \)[/tex]
- [tex]\( 6x + 12y = 126 \)[/tex]
- The first equation here is incorrect. This option is not correct.

d. [tex]\( x + y = 13 \)[/tex]
- [tex]\( 6x + 12y = 126(13) \)[/tex]
- The second equation is incorrect due to the multiplication mistake.

Thus, none of the provided options are entirely correct by themselves. However, if we correct option (a) to include the missing equation, it then matches our model:

[tex]\[ x + y = 13 \][/tex]
[tex]\[ 6x + 12y = 126 \][/tex]

So, the correct system of equations is:

[tex]\[ x + y = 13 \][/tex]
[tex]\[ 6x + 12y = 126 \][/tex]

In this context, option (a) becomes correct if completed with the second equation. This matches the given solution based on the combination of the two key requirements.
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