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Determine the equivalent system for the given system of equations:

5x − 3y = 6
x + y = 2

A. 5x − 3y = 6
6x − 2y = 8
B. 5x − 3y = 6
2x + 2y = 2
C. −5x − 3y = 6
x + y = 2
D. 5x − 3y = 6
6x − 2y = 2

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Answer:

C is the only one equivalent

Step-by-step explanation:

A. 5x − 3y = 6

6x − 2y = 8

Resolving  the orginial system and this system we have

5x − 3y = 6      → 5x − 3y = 6    → -8y = - 4 → y=2 → x=0

(-5)(x + y = 2)      -5x -5y = -10

A. 5x − 3y = 6         →  5x − 3y = 6   →  -4x = -6 → x=3 → y=3

( -3/2)(6x − 2y = 8)    - 9x + 3y = -12

A is not an equivalent system because the 2nd eq of each one  have a different solution

B. 5x − 3y = 6

2x + 2y = 2

Resolving  the orginial system and this system we have

5x − 3y = 6      → 5x − 3y = 6    → -8y = - 4 → y=2 → x=0

(-5)(x + y = 2)      -5x -5y = -10

B 5x − 3y = 6         →  5x − 3y = 6   →  8x = 9 → x=9/8→ y= -1/8

(3/2)(2x + 2y = 2)       3x + 3y = 3

B is not an equivalent system because the 2nd eq of each one  have a different solution

C. −5x − 3y = 6

x + y = 2

C is the same system of the original

D. 5x − 3y = 6

  6x − 2y = 2

Resolving  the orginial system and this system we have

5x − 3y = 6      → 5x − 3y = 6    → -8y = - 4 → y=2 → x=0

(-5)(x + y = 2)      -5x -5y = -10

D 5x − 3y = 6         →  5x − 3y = 6   →  2x = 2 → x=1→ y= - 1/3

(-3/2)(6x − 2y = 2)      - 3x + 3y = -3

D is not an equivalent system because the 2nd eq of each one  have a different solution

Answer:

A.

Step-by-step explanation:

To find the equivalent system of equations, we first need to find the solutions of the given system, and then, see which option has the same solutions.

So, we solve the system of equations given by elimination, to do so, we multiply the second equation by 3, and then sum both of them:

[tex]\left \{ {{5x-3y=6} \atop {x+y=2}} \right.\\\left \{ {{5x-3y=6} \atop {3x+3y=6}} \right.\\8x=12\\x=\frac{12}{8}=\frac{3}{2}[/tex]

Now, we replace this value in one of the equation to find the other solution:

[tex]x+y=2\\\frac{3}{2}+y=2\\y=2-\frac{3}{2}\\y=\frac{4-3}{2}=\frac{1}{2}[/tex]

Therefore, the solution is [tex](\frac{3}{2};\frac{1}{2})[/tex]

Now, we test these values in each option to find the answer that matches.

Option A.

[tex]\left \{ {{5x-3y=6} \atop {6x-2y=8}} \right.\\\left \{ {{5\frac{3}{2} -3\frac{1}{2} =6} \atop {6\frac{3}{2} -2\frac{1}{2} =8}} \right.\\\left \{ {{\frac{15}{2}-\frac{3}{2}=6} \atop {9-1=8}} \right.\\\\\\\left \{ {{\frac{12}{2} =6} \atop {8=8}} \right.[/tex]

As you can see, the first option matches perfectly, the solutions of the given system are also the solutions of the Option A. Therefore they are equivalent systems of equations.

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