The point (5, -3) is a solution to which system of equations?(1 point)Responses

To determine which system of equations has the point (5, -3) as a solution, we will substitute the x-coordinate (5) and the y-coordinate (-3) into each equation in the systems provided.

Option 1:

1. Equation 1: \( y = \frac{2}{3}x + 6 \)
   Substituting:
   \(-3 = \frac{2}{3}(5) + 6\)
   \(-3 = \frac{10}{3} + 6\)
   \(-3 = \frac{10}{3} + \frac{18}{3} = \frac{28}{3}\) (False)

2. Equation 2: \( 3y + 6x - 18 = 0 \)
   Substituting:
   \( 3(-3) + 6(5) - 18 = 0\)
   \(-9 + 30 - 18 = 0\)
   \(3 = 0\) (False)

Option 2:

1. Equation 1: \( y = x - 8 \)
   Substituting:
   \(-3 = 5 - 8\)
   \(-3 = -3\) (True)

2. Equation 2: \( 2x + y = 17 \)
   Substituting:
   \( 2(5) + (-3) = 17\)
   \( 10 - 3 = 17\)
   \(7 = 17\) (False)

Option 3:

1. Equation 1: \( y = -\frac{1}{2}x + 6 \)
   Substituting:
   \(-3 = -\frac{1}{2}(5) + 6\)
   \(-3 = -\frac{5}{2} + 6\)
   \(-3 = -\frac{5}{2} + \frac{12}{2} = \frac{7}{2}\) (False)

2. Equation 2: \( y = 3x - 1 \)
   Substituting:
   \(-3 = 3(5) - 1\)
   \(-3 = 15 - 1\)
   \(-3 = 14\) (False)

Option 4:

1. Equation 1: \( y = x + 2 \)
   Substituting:
   \(-3 = 5 + 2\)
   \(-3 = 7\) (False)

2. Equation 2: \( y = x + 5 \)
   Substituting:
   \(-3 = 5 + 5\)
   \(-3 = 10\) (False)

Summary:

The only equations where the point (5, -3) works is in Option 2 for the first equation only. Therefore, the point (5, -3) is a solution to the system of equations:

y = x - 8
2x + y = 17 (only the first equation is satisfied; the second one is not)

There are no valid pairs of all equations, since the first is true and the second is false.