Solving Systems of Equations for the Value of 8xy

This article dives into solving a system of linear equations to find the value of 8xy. We will walk through the steps methodically and detail the algebraic manipulations required. Understanding systems of equations and the methods to solve them is crucial in various mathematical and practical applications.

The Given System of Equations

The problem at hand provides us with two equations:

2x3y6 y-3x1

We need to solve these equations simultaneously to find the values of x and y, and then use these values to find the product 8xy.

Step 1: Isolate y in the Second Equation

From the second equation, let's isolate y:

y - 3x 1

y 3x 1

This expression for y can be substituted into the first equation to eliminate one variable and solve for the other.

Step 2: Substitute y into the First Equation

Substituting y from the second equation into the first equation:

2x(3x 1) 6

Expanding and simplifying:

6x2 2x - 6 0

We can solve this quadratic equation using the quadratic formula or by factoring. For brevity, let's use the factoring method:

6x2 2x - 6 0

6x2 6x - 4x - 6 0

6x(x 1) - 4(x 1) 0

(6x - 4)(x 1) 0

Solving for x:

6x - 4 0 or x 1 0

x 2/3 or x -1

Next, we substitute these values back into the expression for y:

y 3x 1

For x 2/3:

y 3(2/3) 1 2 1 3

For x -1:

y 3(-1) 1 -3 1 -2 (This solution is not feasible as it does not satisfy the original equations)

Hence, we accept x 2/3 and y 3.

Step 3: Calculate 8xy

Now, calculate 8xy using the values found:

8xy 8 * (2/3) * 3

8xy 8 * 2 16/3 * 3 16/3 * 3/1 16

Thus, the value of 8xy is 4.

Conclusion

By systematically solving the system of equations and performing the necessary algebraic manipulations, we can determine that the value of 8xy is 4. Understanding these steps is crucial for tackling similar problems in algebra and beyond.