Problem 7 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the ordered triple given is the solution to the system of equations. $$ \begin{array}{c} 6 x-y+3 z=6 \\ 3 x+5 y+2 z=0 \quad \text { and }(3,-3,-5) \\ x+y=0 \end{array} $$

Step-by-Step Solution

Answer

The ordered triple (3, -3, -5) is not a solution to the system.

Step 1: Understand the System of Equations

We have a system of three equations:\[\begin{aligned}&1. \ 6x - y + 3z = 6 \&2. \ 3x + 5y + 2z = 0 \&3. \ x + y = 0\end{aligned}\] and an ordered triple (3, -3, -5) to verify if it satisfies all three equations.

Step 2: Substitute the Triple into the First Equation

Substitute \(x = 3\), \(y = -3\), and \(z = -5\) into the first equation: \[6x - y + 3z = 6\].Calculate: \[\begin{aligned}6(3) - (-3) + 3(-5) &= 18 + 3 - 15 \&= 6\end{aligned}\]Since 6 = 6, the first equation is satisfied.

Step 3: Verify the Second Equation

Substitute \(x = 3\), \(y = -3\), and \(z = -5\) into the second equation: \[3x + 5y + 2z = 0\].Calculate:\[\begin{aligned}3(3) + 5(-3) + 2(-5) &= 9 - 15 - 10 \&= -16\end{aligned}\]Since \(-16 eq 0\), the second equation is not satisfied.

Step 4: Check the Third Equation

Substitute \(x = 3\) and \(y = -3\) into the third equation: \[x + y = 0\].Calculate:\[3 + (-3) = 0\]This is true, so the third equation is satisfied.

Step 5: Conclusion

The ordered triple (3, -3, -5) satisfies the first and third equations but does not satisfy the second equation. Therefore, it is not a solution to the entire system.

Key Concepts

Ordered Triple

In a system of equations, an ordered triple refers to a specific set of values for the variables involved. In this case, the ordered triple is \((x, y, z) = (3, -3, -5)\). The values are assigned to the variables \(x\), \(y\), and \(z\) respectively. Each of these represents a solution candidate that can be plugged into each equation in the system to determine if it satisfies all equations.
Imagine a three-dimensional space defined by the axes \(x\), \(y\), and \(z\). An ordered triple is like a point in that space. It is crucial to ensure that the selected point (ordered triple) makes all parts of the system true. If it does, it is a solution to the system. If not, there are other variables or points to be tested.

Equation Verification

Equation verification involves checking if a given set of values (an ordered triple) satisfies each equation in a system. This process helps you confirm whether the ordered triple is indeed a solution. To verify each equation:

• Plug the values of the ordered triple into each equation.
• Simplify the mathematical expressions.
• Check if both sides of each equation are equal after simplification.

In our system:

• The ordered triple \((3, -3, -5)\) satisfies the first equation: \(6(3) - (-3) + 3(-5) = 6\). The left side simplifies to \(6\), which equals the right side.
• Similarly, it satisfies the third equation: \(3 + (-3) = 0\).
• However, the second equation is not satisfied: \(3(3) + 5(-3) + 2(-5) = -16\), which does not equal \(0\).

Each equation needs verification with the full ordered triple to confirm its validity.

Substitution Method

The substitution method in solving systems of equations involves replacing the variables in one equation with the values or expressions from another equation. Here, we apply substitution by directly replacing \(x\), \(y\), and \(z\) in each of our equations with the values provided by the ordered triple.
The substitution method can be strategic because:

• It allows you to work with simpler expressions.
• You can solve complicated systems step-by-step, focusing on one equation at a time.
• It works well when one set of values is readily available to be substituted, reducing the computation needed for each step.

In our system of equations, substituting \((3, -3, -5)\) into each equation is a straightforward step to see if it satisfies all equations at once. The method ensures we systematically check each equation, avoiding oversight errors.