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Elimination was used to solve a system of equations. One of the intermediate steps led to the equation [tex]$7x = 21$[/tex]. Which of the following systems could have led to this equation?

A. [tex]7x + 2y = 21[/tex]
[tex]-7x - 2y = 21[/tex]

B. [tex]8x - y = 15[/tex]
[tex]x + y = -6[/tex]

C. [tex]3x + 2y = 7[/tex]
[tex]2x - y = 7[/tex]

D. [tex]4x + 3y = 24[/tex]
[tex]-3x - 3y = 3[/tex]

Sagot :

GinnyAnsweridn
To find which systems of equations could have led to the equation [tex]\(7x = 21\)[/tex] through the elimination method, we need to analyze each of the given systems and check if eliminating variables could simplify to [tex]\(7x = 21\)[/tex].

1. System: [tex]\(7x + 2y = 21\)[/tex]
- If [tex]\(2y = 0\)[/tex], then we have [tex]\(7x = 21\)[/tex].
- This is already in the desired form, so this system works.

2. System: [tex]\(-7x - 2y = 21\)[/tex]
- Rearranging terms, we can write [tex]\(7x + 2y = -21\)[/tex].
- Since [tex]\(7x = 21\)[/tex] would require the coefficients of [tex]\(x\)[/tex] to be positive, this system leads to [tex]\(x\)[/tex] being negative, not consistent with the desired equation.

3. System: [tex]\(8x - y = 15\)[/tex] and [tex]\(x + y = -6\)[/tex]
- Adding the two equations: [tex]\(8x - y + x + y = 15 - 6\)[/tex]
- This simplifies to [tex]\(9x = 9\)[/tex] and [tex]\(x = 1\)[/tex], not leading to [tex]\(7x = 21\)[/tex].

4. System: [tex]\(3x + 2y = 7\)[/tex] and [tex]\(2x - y = 7\)[/tex]
- Multiply the second equation by 2: [tex]\(4x - 2y = 14\)[/tex]
- Adding to the first equation: [tex]\(3x + 2y + 4x - 2y = 7 + 14\)[/tex]
- This simplifies to [tex]\(7x = 21\)[/tex].

5. System: [tex]\(4x + 3y = 24\)[/tex]
- This is a single equation. We cannot eliminate a variable to achieve [tex]\(7x = 21\)[/tex] with just one equation.

6. System: [tex]\(-3x - 3y = 3\)[/tex]
- Rearranging terms, we have [tex]\(3x + 3y = -3\)[/tex]
- This does not lead to [tex]\(7x = 21\)[/tex] as the coefficients and constants do not align with elimination producing that result.

Next, because [tex]\(7x = 21\)[/tex] simplifies to [tex]\(x = 3\)[/tex], it is consistent with both [tex]\(7x + 2y = 21\)[/tex] where [tex]\(2y = 0\; (y\)[/tex] can be any value), and the pair [tex]\(3x + 2y = 7\)[/tex] and [tex]\(4 x + 3 y = 24\)[/tex].

Therefore, the systems that could have led to the equation [tex]\(7x = 21\)[/tex] are:
1. [tex]\(7 x + 2 y = 21\)[/tex]
2. [tex]\(3 x + 2 y = 7\)[/tex] and [tex]\(4 x + 3 y = 24\)[/tex]

Hence, the valid system numbers are:
[tex]\[ [1, 3, 5] \][/tex]
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