Problem 8 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the ordered triple given is the solution to the system of equations. $$ \begin{aligned} 6 x-7 y+z=& 2 \\ -x-y+3 z &=4 \quad \text { and }(4,2,-6) \\ 2 x+y-z &=1 \end{aligned} $$

Step-by-Step Solution

Answer

The ordered triple \((4, 2, -6)\) is not a solution to the system of equations.

Step 1: Substitute the values into the first equation

The first equation given is \(6x - 7y + z = 2\). Substitute \(x = 4\), \(y = 2\), and \(z = -6\) into this equation to check if it holds true. So, calculate \(6(4) - 7(2) + (-6)\). Calculate to get:\(24 - 14 - 6 = 4\). This equation does not satisfy with 4 as the result, instead of 2.

Step 2: Check the second equation

Next, we check the second equation \(-x - y + 3z = 4\) using the same values.Substitute \(x = 4\), \(y = 2\), and \(z = -6\) into the equation:\(-4 - 2 + 3(-6)\).Calculate to get:\(-4 - 2 - 18 = -24\).This does not satisfy the equation because -24 is not equal to 4.

Step 3: Check the third equation

The last equation is \(2x + y - z = 1\). We need to substitute \(x = 4\), \(y = 2\), and \(z = -6\) into this equation.Calculate \(2(4) + 2 - (-6)\).Calculate to get:\(8 + 2 + 6 = 16\).This does not satisfy the equation since 16 is not equal to 1.

Step 4: Conclusion: Verify the solution

Since none of the three simultaneous equations hold true with the ordered triple \((4, 2, -6)\), it is not a solution to the system.

Key Concepts

Ordered Triples

An ordered triple is a set of three values, each representing a solution to the variables in a three-variable system, often written as \((x, y, z)\). This notation shows the specific values for each variable in a certain order. In the context of solving systems of equations, ordered triples are used to determine if the given values satisfy all equations in the system concurrently.

• Ordered triples must coordinate with three variables: \(x\), \(y\), and \(z\).
• Each number in the triple corresponds to one variable in the system.
• Verifying if a triple is a solution involves checking if these values satisfy all equations simultaneously.

Suppose an ordered triple solves a system of equations. In that case, it means that when substituted into each equation, each side of the equation remains balanced or equal. This unique intersection of all equations means the ordered triple is a solution to the system.

Equation Substitution

Equation substitution is the process of replacing the variables within an equation with given numerical values. By doing so, we test whether these particular values satisfy an equation. It's a method used to verify if a set of values, like an ordered triple, belongs to the solution set of the system of equations.To substitute effectively, follow these steps:

• Identify the variables in the equation and the corresponding values from the ordered triple.
• Replace each variable with the appropriate value from the triple.
• Calculate the resulting expression to see if both sides of the equation are equal.

Using substitution gives clarity about whether the ordered triple satisfies the equation or not. In our original exercise, substituting the values \((4, 2, -6)\) didn't satisfy any of the equations, concluding that the ordered triple is not a solution.

Solution Verification

Solution verification is the last step in problem-solving when dealing with systems of equations. It involves checking whether the given ordered triple makes all the equations true.To properly verify a solution:

• Substitute the values from the ordered triple into each individual equation of the system.
• After substitution, calculate the result for each equation.
• Check if the results match the corresponding constants on the right-hand side of the equations.

If all equations are true when using the ordered triple, this indicates it is indeed the solution. In the exercise, the triple \((4, 2, -6)\) was validated against every equation, and none was true. Therefore, the triple does not solve the system, highlighting how solution verification helps confirm or reject potential solutions.