Math  /  Algebra

Question14 Which system of equations has the same solution as the system below? 2x+2y=163xy=4\begin{array}{c} 2 x+2 y=16 \\ 3 x-y=4 \end{array} (1) 2x+2y=166x2y=4\begin{array}{l} 2 x+2 y=16 \\ 6 x-2 y=4 \end{array} (3) x+y=16x+y=16 3xy=43 x-y=4 (2) 2x+2y=162 x+2 y=16  (4) 6x+6y=486x+2y=8\text { (4) } \begin{aligned} 6 x+6 y & =48 \\ 6 x+2 y & =8 \end{aligned} 6x2y=86 x-2 y=8

Studdy Solution

STEP 1

1. We need to determine which system of equations has the same solution as the given system.
2. Two systems of equations have the same solution if they are equivalent, meaning they represent the same lines or planes in a coordinate system.

STEP 2

1. Analyze the given system of equations.
2. Analyze each option to determine if it is equivalent to the given system.
3. Check if the solutions of the given system and each option match.

STEP 3

Analyze the given system of equations:
The given system is: 2x+2y=163xy=4\begin{array}{c} 2x + 2y = 16 \\ 3x - y = 4 \end{array}
We can simplify the first equation by dividing every term by 2: x+y=8x + y = 8
The system becomes: x+y=83xy=4\begin{array}{c} x + y = 8 \\ 3x - y = 4 \end{array}

STEP 4

Analyze each option:
Option (1): 2x+2y=166x2y=4\begin{array}{c} 2x + 2y = 16 \\ 6x - 2y = 4 \end{array}
Simplify the second equation by dividing every term by 2: 3xy=23x - y = 2
This is not equivalent to the original system because the second equation is different.

STEP 5

Option (2): 2x+2y=163xy=4\begin{array}{c} 2x + 2y = 16 \\ 3x - y = 4 \end{array}
This is identical to the original system, so it is equivalent.

STEP 6

Option (3): x+y=163xy=4\begin{array}{c} x + y = 16 \\ 3x - y = 4 \end{array}
The first equation is different from the simplified form of the original system, so it is not equivalent.

STEP 7

Option (4): 6x+6y=486x2y=8\begin{array}{c} 6x + 6y = 48 \\ 6x - 2y = 8 \end{array}
Simplify the first equation by dividing every term by 6: x+y=8x + y = 8
Simplify the second equation by dividing every term by 2: 3xy=43x - y = 4
This matches the simplified form of the original system, so it is equivalent.

STEP 8

The systems that are equivalent to the original system are Option (2) and Option (4).
The systems that have the same solution as the given system are Option (2) and Option (4).

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