Solving a System of Equations for x, y, z, and w

This article explores the solution of a system of equations involving four variables: x, y, z, and w. The given equations are:

x * y 14 x * z 16 z - w 10 y * w 15

We will solve this system step-by-step using substitution and algebraic manipulation.

Step 1: Expressing y and z in terms of x

From the first equation:

x * y 14

y 14 - x

From the second equation:

x * z 16

z 16 - x

Step 2: Substituting z into the third equation

Substituting z 16 - x into the third equation:

z - w 10

16 - x - w 10

16 - x - w 10

-w x - 6

w 6 - x

Step 3: Substituting y and w into the fourth equation

Now substituting y 14 - x and w 6 - x into the fourth equation:

y * w 15

(14 - x) * (6 - x) 15

20 - 2x - 84 6x 15

20 - 68 4x 15

-48 4x 15

4x 63

x 15.75 / 4 3.9375

Step 4: Calculating the values of y, z, and w

Substituting x 3.9375 back into the original equations:

y 14 - 3.9375 10.0625

z 16 - 3.9375 12.0625

w 6 - 3.9375 2.0625

Summary of Values

x 3.9375

y 10.0625

z 12.0625

w 2.0625

Thus, the final values are:

x 3.9375 y 10.0625 z 12.0625 w 2.0625

The detailed solution and method for solving the system of equations involving the variables x, y, z, and w demonstrate the application of algebraic techniques and strategic substitution, highlighting the importance of systematic problem-solving in mathematics.