Question

Multiple Choice If a system of dependent equations containing three variables has the general solution \(\\{(x, y, z) \mid x=-z+4, y=-2 z+5, z\) is any real number \(\\}\) then _____ is one of the infinite number of solutions of the system. (a) (1,-1,3) (b) (0,4,5) (c) (4,-3,0) (d) (-1,5,7)

Short answer

The correct option is (a): (1, -1, 3).

Step-by-step solution

Rewrite the general solution

The general solution of the system is given as \(x=-z+4, y=-2z+5, z \). This means for any real number value of \(z\), we can find corresponding values for \(x\) and \(y\) using the equations.

Test option (a)

For (1, -1, 3), let’s set \(z = 3\). Then, calculate \(x\) and \(y\): \[x = -z + 4 = -3 + 4 = 1\] \[y = -2z + 5 = -2(3) + 5 = -6 + 5 = -1\] \Hence, (1, -1, 3) is a solution.\

Test option (b)

For (0, 4, 5), let’s set \(z = 5\). Then, calculate \(x\) and \(y\): \[x = -z + 4 = -5 + 4 = -1\] \[y = -2z + 5 = -2(5) + 5 = -10 + 5 = -5\] \Therefore, (0, 4, 5) is not a solution.\

Test option (c)

For (4, -3, 0), let’s set \(z = 0\). Then, calculate \(x\) and \(y\): \[x = -z + 4 = -0 + 4 = 4\] \[y = -2z + 5 = -2(0) + 5 = 5\] \Therefore, (4, -3, 0) is not a solution.\

Test option (d)

For (-1, 5, 7), let’s set \(z = 7\). Then, calculate \(x\) and \(y\): \[x = -z + 4 = -7 + 4 = -3\] \[y = -2z + 5 = -2(7) + 5 = -14 + 5 = -9\] \Therefore, (-1, 5, 7) is not a solution.\

Conclusion

Out of the given options, only (1, -1, 3) satisfies the general solution of the system \(x=-z+4, y=-2z+5, z\).

Key concepts

Dependent Equations

Dependent equations are a set of equations where at least one equation can be derived from the others. They represent the same geometric object in space.
In simpler terms, dependent equations have a special relationship where solving one equation automatically solves the others.
For example, in the provided problem, the equations for \(x\), \(y\), and \(z\) are all interlinked. By knowing the value of \(z\), we can always determine the values of \(x\) and \(y\).
Dependent equations are crucial when working with systems having infinite solutions. Each solution is not isolated but part of a family of solutions.
Think of a straight line in a 2D plane. Any point on that line is a solution to the equations representing that line. The same applies to higher dimensions.
Remember, recognizing dependent equations helps in understanding their nature and quickly finding the general solution.

General Solution

The general solution of a system of equations provides a way to express all potential solutions. For dependent equations, it means describing a family of solutions with a parameter.
In our exercise, the general solution is given as \(x = -z + 4, y = -2z + 5, z\), where \(z\) is any real number.
This tells us that given any value of \(z\), we can always find the corresponding \(x\) and \(y\).

Here's a breakdown on how to find the general solution:

• Choose the independent variable (variable you can freely choose), which in this case is \(z\).
• Express the other variables in terms of the chosen variable.

The general solution is important because it captures all possible solutions to the system, offering a comprehensive understanding.
Instead of finding isolated solutions, you have a full description that includes every potential combination of \(x, y, z\).

Three Variables

When dealing with systems of equations in three variables, you're working in three-dimensional space. Each variable represents a dimension.
Visualize each equation as a plane intersecting in space. Dependent planes will intersect along a line or coincide entirely.
For example, the given system \(x = -z + 4, y = -2z + 5\) can be visualized as equations of planes intersecting along a line.
Here's how you approach such a problem:

• Identify the independent variable (\(z\) in this case).
• Substitute values for the independent variable to find corresponding values of the other variables.
• This process creates points in \(x, y, z\) space that lie on the intersection line of the planes.

Understanding the behavior of equations in three dimensions helps in visualizing how solutions fit geometrically.
Instead of intersecting at a point, dependent systems intersect along a line, demonstrating why they offer infinite solutions.

Breaking down problems step by step and visualizing the relationship between equations and their geometric representations can make handling such systems easier.