The following equations are given Equation #1 3x+z+y=8 Equation #2 5y-x=-7 Equation #3 3z+2x-2y=15 Equatio

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1. Using a system of equations, we have that:

   a) It is not possible to solve for any of the variables using only Equation #1 and Equation #2, as there are only 2 equations for 3 variables.

   b) It is possible to solve for any of the variables using only Equation #1, Equation #2, and Equation #3, as there are 3 equations for 3 variables. The solutions are: x = 12, y = 1, z = -29.

   c) These solutions do not hold for equation 4, meaning that equation 4 is inconsistent for the system.

   What is a system of equations?

   A system of equations is when two or more variables are related, and equations are built to find the values of each variable.

   To solve a system, the number of equations has to be at least the same as the number of variables, hence in item a, it is not possible to solve for any of the variables using only Equation #1 and Equation #2, as there are only 2 equations for 3 variables.

   In item b, we can solve. From the second equation, we have that:

   x = 5y + 7.

   From the first equation, we have that:

   z = 8 – 3x – y

   z = 8 – 3(5y + 7) – y

   z = 8 – 15y – 21 – y

   z = -13 – 16y

   From the third equation, we can replace and solve for y.

   3z + 2x – 2y = 15.

   3(-13 – 16y) + 2(5y + 7) – 2y = 15

   -39 – 48y + 10y + 14 – 2y = 15.

   -40y – 25 = 15

   40y = 40.

   y = 1.

   Then the solutions for x and z are:

   • x = 5y + 7 = 12.

   • z = -13 – 16y = -29.

   Replacing the solutions in equation 4, we have that:

   4x + 5y – 2z = -3.

   4(12) + 5(1) – 2(-29) = -3, which is false.

   These solutions do not hold for equation 4, meaning that equation 4 is inconsistent for the system.

   More can be learned about a system of equations at https://brainly.com/question/24342899

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