Understanding the Equation 3x 2y 8z - 5 0

The equation 3x 2y 8z - 5 0 is a linear equation in three-dimensional space. This equation represents a plane and has infinitely many solutions. Let's delve into how to solve and understand this equation.

Solving for One Variable

To solve for one variable, we can express it in terms of the other two. Here's how to do it:

Solve for z:
Solving for z: 8z -3x - 2y - 5
boxed{z frac{-3x - 2y - 5}{8}} Solve for y:
Solving for y: 2y -3x - 8z - 5
boxed{y frac{-3x - 8z - 5}{2}} Solve for x:
Solving for x: 3x -2y - 8z - 5
boxed{x frac{-2y - 8z - 5}{3}}

These expressions allow you to find one variable given the values of the other two. For a specific solution, you can choose values for two variables and solve for the third.

Example Solutions

For example, let's find a specific solution by choosing x 0 and y 0.

Substitute x 0 and y 0 into the equation: 3(0) 2(0) 8z - 5 0
8z 5
boxed{z frac{5}{8}} So one solution is (0, 0, 5/8).

By varying the values of x and y, you can find more solutions.

Further Explanation

This process may seem complex, but it is quite straightforward when broken down step by step. Solving an equation with one unknown involves finding a value that satisfies the equation. However, solving multiple unknowns requires multiple equations. For instance:

3x 15
Here, x 5. 3x 2y 23
This represents a line, and there are infinitely many pairs (x, y). 3x 2y 8z 5
This is a plane, and there are infinitely many triples of values (x, y, z).

Therefore, the equation 3x 2y 8z - 5 0, which represents a plane, cannot be solved to find an exact unique solution because it has three unknowns (x, y, z) and requires three equations to find a specific solution.

However, you can obtain arbitrary solutions by expressing one variable in terms of the other two and selecting values for those variables. Here's an example:

Assume x a and y b. Substitute into the equation to solve for z: 3a 2b 8z - 5 0
8z 5 - 3a - 2b
boxed{z frac{5 - 3a - 2b}{8}} Vary the values of a and b to find different z values.

This method provides an infinite number of solutions based on the chosen values of a and b.

Conclusion

The equation 3x 2y 8z - 5 0 is a linear equation that represents a plane in three-dimensional space. It has infinitely many solutions, and finding specific solutions involves expressing one variable in terms of the others or selecting specific values for the variables. This detailed understanding helps in solving such equations and finding suitable solutions.