Problem 41 - Intermediate Algebra

Question

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. $$ \left\\{\begin{array}{l} 2 x+2 y-z=2 \\ x+3 z-24=0 \\ y=7-4 z \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = 12\), \(y = -9\), and \(z = 4\). The system is consistent and independent.

Step 1: Analyze the System of Equations

The given system of equations is:\[\begin{aligned}1. & \quad 2x + 2y - z = 2 \2. & \quad x + 3z = 24 \3. & \quad y = 7 - 4z\end{aligned}\]We will solve these equations by substitution and elimination methods.

Step 2: Express Variables in Terms of z

Using Equation 3, we have already expressed \(y\) in terms of \(z\):\[y = 7 - 4z\]From Equation 2, express \(x\) in terms of \(z\):\[x = 24 - 3z\]

Step 3: Substitute for x and y in Equation 1

Substitute \(x = 24 - 3z\) and \(y = 7 - 4z\) into Equation 1:\[2(24 - 3z) + 2(7 - 4z) - z = 2\]

Step 4: Simplify and Solve for z

Simplify the equation:\[48 - 6z + 14 - 8z - z = 2\]Combine like terms:\[62 - 15z = 2\]Rearrange to solve for \(z\):\[15z = 60\]\[z = 4\]

Step 5: Substitute z back to find x and y

Use \(z = 4\) in the expressions for \(x\) and \(y\):\[y = 7 - 4(4) = 7 - 16 = -9\]\[x = 24 - 3(4) = 24 - 12 = 12\]

Step 6: Verify the Solution

Substitute \(x = 12\), \(y = -9\), and \(z = 4\) back into the original equations to verify:- Equation 1: \(2(12) + 2(-9) - 4 = 2\), which simplifies to \(24 - 18 - 4 = 2\), true.- Equation 2: \(12 + 3(4) = 24\), which simplifies to \(12 + 12 = 24\), true.- Equation 3 is already satisfied by substitution.All conditions are satisfied, so the solution is correct.

Key Concepts

Substitution Method

The substitution method is a powerful technique used for solving systems of equations by expressing variables in terms of each other. In our exercise, this involved expressing both \(x\) and \(y\) in terms of \(z\).
By doing this:

• We simplified a system of three equations into a single equation that only depended on \(z\).
• From equation 3, we directly found that \(y = 7 - 4z\).
• From equation 2, rearranging terms allowed us to express \(x = 24 - 3z\).

This approach reduces complexity and lets us focus on a single variable first, making it easier to solve for the remaining variables once \(z\) was determined.

Elimination Method

The elimination method involves adding or subtracting equations to eliminate one of the variables. In this exercise, we mainly used substitution to first express other equations in terms of a single variable. However, elimination can be extremely useful:

• When we have equations like our first example (steps involving x and y), manipulating the equations can completely remove a variable.
• This "combines" the original equations into a new one with fewer variables.

Despite not being predominantly used here, elimination can significantly simplify problems or act as a complementary strategy alongside substitution.

Dependent Systems

A dependent system occurs when the equations represent the same line or plane, meaning they have infinitely many solutions. In our exercise:

• Each equation provided a unique constraint on the solution.
• The system was neither dependent nor did it have infinitely many solutions.

Understanding whether a system is dependent helps in realizing that you're dealing with a single geometric entity expressed in different ways, which results in redundancy.

Inconsistent Systems

Inconsistent systems do not have a solution because the equations contradict each other. In the given problem:

• Each equation had to work together to meet all conditions, with no equation directly opposing another.
• Had there been any logical contradiction, it would signal an inconsistent system.

Recognizing inconsistency early can save effort and redirect strategies toward identifying the contradiction instead of searching for non-existent solutions.