11.2 Systems of linear equations: three variables  (Page 4/8)

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Does the generic solution to a dependent system always have to be written in terms of $\text{\hspace{0.17em}}x?$

No, you can write the generic solution in terms of any of the variables, but it is common to write it in terms of x and if needed $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y.$

Solve the following system.

Infinite number of solutions of the form $\text{\hspace{0.17em}}\left(x,4x-11,-5x+18\right).\text{\hspace{0.17em}}$

Access these online resources for additional instruction and practice with systems of equations in three variables.

Key concepts

• A solution set is an ordered triple $\text{\hspace{0.17em}}\left\{\left(x,y,z\right)\right\}\text{\hspace{0.17em}}$ that represents the intersection of three planes in space. See
• A system of three equations in three variables can be solved by using a series of steps that forces a variable to be eliminated. The steps include interchanging the order of equations, multiplying both sides of an equation by a nonzero constant, and adding a nonzero multiple of one equation to another equation. See [link] .
• Systems of three equations in three variables are useful for solving many different types of real-world problems. See [link] .
• A system of equations in three variables is inconsistent if no solution exists. After performing elimination operations, the result is a contradiction. See [link] .
• Systems of equations in three variables that are inconsistent could result from three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location.
• A system of equations in three variables is dependent if it has an infinite number of solutions. After performing elimination operations, the result is an identity. See [link] .
• Systems of equations in three variables that are dependent could result from three identical planes, three planes intersecting at a line, or two identical planes that intersect the third on a line.

Verbal

Can a linear system of three equations have exactly two solutions? Explain why or why not

No, there can be only one, zero, or infinitely many solutions.

If a given ordered triple solves the system of equations, is that solution unique? If so, explain why. If not, give an example where it is not unique.

If a given ordered triple does not solve the system of equations, is there no solution? If so, explain why. If not, give an example.

Not necessarily. There could be zero, one, or infinitely many solutions. For example, $\text{\hspace{0.17em}}\left(0,0,0\right)\text{\hspace{0.17em}}$ is not a solution to the system below, but that does not mean that it has no solution.

Using the method of addition, is there only one way to solve the system?

Can you explain whether there can be only one method to solve a linear system of equations? If yes, give an example of such a system of equations. If not, explain why not.

Every system of equations can be solved graphically, by substitution, and by addition. However, systems of three equations become very complex to solve graphically so other methods are usually preferable.

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