Tina wrote the equations 3 x minus y = 9 and 4 x + y = 5. What can Tina conclude about the solution to this system of equations?

Question

(–3, 2) is a solution to the system of linear equations.
(–2, 3) is a solution to the system of linear equations.
(2, –3) is a solution to the system of linear equations.
(3, –2) is a solution to the system of linear equations.

Answer 1

To determine which of the given ordered pairs is a solution to the system of equations provided by Tina, we need to substitute each pair into both equations and check if they satisfy both.

The equations are:

1. \(3x - y = 9\)
2. \(4x + y = 5\)

We will test each of the given points:

1. For (–3, 2):

   • Equation 1: \(3(-3) - 2 = -9 - 2 = -11 \quad (\text{not } 9)\)
   • Equation 2: \(4(-3) + 2 = -12 + 2 = -10 \quad (\text{not } 5)\)

2. For (–2, 3):

   • Equation 1: \(3(-2) - 3 = -6 - 3 = -9 \quad (\text{not } 9)\)
   • Equation 2: \(4(-2) + 3 = -8 + 3 = -5 \quad (\text{not } 5)\)

3. For (2, –3):

   • Equation 1: \(3(2) - (-3) = 6 + 3 = 9 \quad (\text{yes } 9)\)
   • Equation 2: \(4(2) + (-3) = 8 - 3 = 5 \quad (\text{yes } 5)\)

4. For (3, –2):

   • Equation 1: \(3(3) - (-2) = 9 + 2 = 11 \quad (\text{not } 9)\)
   • Equation 2: \(4(3) + (-2) = 12 - 2 = 10 \quad (\text{not } 5)\)

From the testing, only the point (2, –3) satisfies both equations:

1. \(3(2) - (-3) = 9\)
2. \(4(2) + (-3) = 5\)

Therefore, Tina can conclude that (2, –3) is a solution to the system of linear equations.