Problem 8 - A Graphical Approach to College Algebra

Question

If the equations are dependent, write the solution set in terms of the variable \(z\). (Hint: In Exercises 3\begin{array}{l} 2 x+y+z=9 \\ -x-y+z=1 \\ 3 x-y+z=9 \end{array}

Step-by-Step Solution

Answer

The solution set is \((x, y, z) = (10 - 2z, z + 5, z)\), where \(z\) is a free parameter.

Step 1: Write the System of Equations

The given system of equations is:\[ \begin{align*} 2x + y + z &= 9, \ -x - y + z &= 1, \ 3x - y + z &= 9. \end{align*} \] Our goal is to find if the equations are dependent and express the solution set in terms of the variable \(z\).

Step 2: Add Equations 1 and 2

Add the first equation \(2x + y + z = 9\) and the second equation \(-x - y + z = 1\):\[ \begin{align*} (2x + y + z) + (-x - y + z) &= 9 + 1, \ x + 2z &= 10. \end{align*} \] This is our first simplified equation.

Step 3: Add Equations 2 and 3

Add the second equation \(-x - y + z = 1\) and the third equation \(3x - y + z = 9\):\[ \begin{align*} (-x - y + z) + (3x - y + z) &= 1 + 9, \ 2x - 2y + 2z &= 10. \end{align*} \] Simplify to get \(x - y + z = 5\). This becomes our second simplified equation.

Step 4: Express Variables in Terms of z

From the two simplified equations, solve for \(x\) and \(y\) in terms of \(z\):From \(x + 2z = 10\):\[ x = 10 - 2z.\]From \(x - y + z = 5\) and substituting \(x\):\[ (10 - 2z) - y + z = 5, \ -y + z = 5 - 10 + 2z, \ -y + z = -5 + 2z, \ y = z + 5. \]Thus, \(x = 10 - 2z\), \(y = z + 5\).

Step 5: Solution Set

Since the equations express the same relationship:The solution set is given by: \[(x, y, z) = (10 - 2z, z + 5, z),\] where \(z\) is a free parameter. This indicates a dependent system with infinitely many solutions.

Key Concepts

Solution Set

A solution set refers to the collection of all possible solutions that satisfy the given system of equations. In our case, a solution set consists of ordered pairs or triples of numbers that make the system of equations true. For the discussed problem, since we have three variables—\(x\), \(y\), and \(z\)— the solution set is expressed in terms of the variable \(z\). This means \(z\) is a free parameter that can take any real value, and the corresponding \(x\) and \(y\) values will adjust accordingly to satisfy all the equations in the system.Understanding how to express one variable in terms of another is key, especially when trying to capture all possible solutions in an efficient manner. This is crucial for variables that are not explicitly given a specific value in the equations.

System of Equations

A system of equations is a set of two or more equations with the same variables. The goal is often to find values for the variables that satisfy all equations simultaneously. In this exercise, we are dealing with the following system:

• \(2x + y + z = 9\)
• \(-x - y + z = 1\)
• \(3x - y + z = 9\)

A productive approach is to simplify the system by combining equations to eliminate variables, making it easier to solve.Simplifying systems of equations helps to identify relationships between the variables, which is particularly useful when the equations are dependent, as we can see by the solutions for \(x\) and \(y\) in terms of \(z\). Understanding these relationships allows for expressing one variable as a function of others, which aids in finding the complete solution set.

Infinitely Many Solutions

Infinitely many solutions occur in a system of equations when the equations are not independent, meaning they convey the same relationship using different forms or multiples. This is often seen when you end up having dependent equations.Dependent equations don't just hint at overlapping solutions; they essentially describe the same solution space. In our problem, the equations were simplified to show that all can be satisfied with the variables expressed as functions of \(z\):

• \(x = 10 - 2z\)
• \(y = z + 5\)
• \(z\) remains a free parameter

This indicates that no single solution exists but rather a continuum of solutions, where \(z\) can be any real number. Hence, the system is described as having infinitely many solutions, leading to a line or plane of solutions rather than a point on a graph.A dependent system like this shows how multiple initial equations boil down to a single underlying constraint represented in a versatile form.