Problem 7 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Insert the Ordered Triple

The ordered triple given in the problem is \((x, y, z) = (3, -3, -5)\). Insert these values into each equation to check if they satisfy all three.

Step 2: Substitute in the First Equation

Substitute \(x = 3\), \(y = -3\), and \(z = -5\) into the first equation \(6x - y + 3z = 6\). The calculation is as follows: \[6(3) - (-3) + 3(-5) = 6\] \[18 + 3 - 15 = 6\] The equation holds true, as both sides equal 6.

Step 3: Substitute in the Second Equation

Now substitute the same values into the second equation \(3x + 5y + 2z = 0\). The calculation is: \[3(3) + 5(-3) + 2(-5) = 0\] \[9 - 15 - 10 = -16\] Both sides are not equal, hence the ordered triple does not satisfy this equation.

Step 4: Substitute in the Third Equation

Finally, substitute into the third equation \(x + y = 0\): \[3 + (-3) = 0\] This equation holds true, as both sides equal 0.

Step 5: Conclusion

Since the ordered triple does not satisfy the second equation, \((3, -3, -5)\) is not a solution to the system of equations.

Key Concepts

Ordered Triples

In mathematics, especially when dealing with systems of equations, the concept of "ordered triples" is important. An ordered triple is a set of three numbers, typically written in the form \((x, y, z)\), where each element corresponds to one of the variables in a system of three equations.

• The ordered triple specifies the potential solution, telling us the values that might satisfy all involved equations simultaneously.
• To verify if an ordered triple is indeed a solution for a system of equations, it must fulfill each equation.

For example, in our exercise, the ordered triple is \((3, -3, -5)\). This means that for each equation in the system, we plug \(x = 3\), \(y = -3\), and \(z = -5\) into the equations to check for consistency.
The idea is that when these values make both sides of the equation equal, the triple can be considered a valid solution for that specific equation.

Substitution Method

The substitution method is a strategy used to solve systems of equations, particularly helpful when you have multiple equations and multiple variables.
This method involves taking one of the equations and solving it for one variable, then substituting that expression in the other equations to reduce the number of variables.

• In our case, however, since we don't need to find a new solution, but to verify an existing triple, we still apply a form of substitution.
• We directly substitute the values provided in the ordered triple into each equation.

For example:

• In the first equation, we substitute \(x = 3\), \(y = -3\), and \(z = -5\), resulting in \[6(3) - (-3) + 3(-5) = 6\].
• This simplifies to \[18 + 3 - 15 = 6\], verifying that the triple satisfies the first equation.

This substitution method helps us determine whether the given values satisfy each specific equation in the system.

Solution Verification

Solution verification is crucial in confirming if a proposed solution is correct for a system of equations.
It involves plugging the proposed values into the equations to ensure that all expressions hold true simultaneously.Here’s a step-by-step breakdown of the process:

• Insert the Ordered Triple: Start by taking the ordered triple provided, in this case, \((3, -3, -5)\), and substitute the values into the respective variables of each equation.
• Check each Equation: For each equation, calculate the left-hand side using the substituted values and compare it to the right-hand side.
  • If they are equal, the ordered triple satisfies that equation.
  • If not, as in our second equation, \(3(3) + 5(-3) + 2(-5) eq 0\), the proposed triple is not a valid solution for the entire system.

By going through each equation, you ensure that the proposed solution is consistent across the whole system.
Solution verification acts as the final check and confirms mathematical accuracy, ensuring the ordered triple fits within the rules established by the equations.