Problem 8 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Substitute Triple into First Equation

Substitute the values from the ordered triple \((x, y, z) = (4, 2, -6)\) into the first equation: \(6x - 7y + z = 2\). Calculate: \(6(4) - 7(2) + (-6)\).

Step 2: Calculate the First Equation

Solve the substituted equation: \(6(4) - 7(2) + (-6) = 24 - 14 - 6 = 4\). Since \(4 eq 2\), the ordered triple does not satisfy the first equation.

Step 3: Substitute Triple into Second Equation

Substitute \((x, y, z) = (4, 2, -6)\) into the second equation: \(-x - y + 3z = 4\). Calculate: \(-4 - 2 + 3(-6)\).

Step 4: Calculate the Second Equation

Solve the substituted equation: \(-4 - 2 - 18 = -24\). Since \(-24 eq 4\), the ordered triple does not satisfy the second equation.

Step 5: Substitute Triple into Third Equation

Substitute \((x, y, z) = (4, 2, -6)\) into the third equation: \(2x + y - z = 1\). Calculate: \(2(4) + 2 - (-6)\).

Step 6: Calculate the Third Equation

Solve the substituted equation: \(8 + 2 + 6 = 16\). Since \(16 eq 1\), the ordered triple does not satisfy the third equation.

Step 7: Conclusion

Since the ordered triple does not satisfy any of the three equations, it is not a solution to the system of equations.

Key Concepts

Ordered Triples

An ordered triple refers to a set of three numbers, typically written as \(x, y, z\). Unlike ordered pairs, which describe points in two-dimensional space, ordered triples represent points in three-dimensional space.
In the context of systems of equations, an ordered triple is a solution that simultaneously satisfies all equations within the system. Each number in the triple corresponds to a particular variable in the system.
For instance, when examining if \(4, 2, -6\) is the solution to a system of equations, we substitute \(x = 4, y = 2,\) and \(z = -6\) into each equation. If all equations are true when using these values, then the ordered triple is indeed the solution.

Substitution Method

The substitution method is a technique used to find solutions for systems of equations. It involves replacing one of the variables with its equivalent expression derived from another equation.
This method is particularly useful for solving systems that include two or more equations. When dealing with ordered triples, you'll have three equations to substitute into and verify.
For our exercise, we substituted the values \(x = 4, y = 2, z = -6\) into each of the given equations in the system to check if they hold true. Through substitution, we test if the given ordered triple, in this instance \(4, 2, -6\), satisfies all involved equations.

Solution to System

A solution to a system of equations is a set of values for the variables that makes all of the equations true simultaneously.
In other words, it is the intersection point where all equations meet in the coordinate space, which can be visualized as lines, planes, or more complex surfaces depending on the number of variables.
In this example, \(4, 2, -6\) was checked to see if it was a solution. Since it didn't satisfy any of the equations, it was concluded not to be a solution. An ordered triple that solves the system would need to make every substituted equation valid.

Linear Equations

Linear equations describe straight lines when graphed on a coordinate plane and take the form \ ax + by + cz = d\ for three variables.
Each equation in a system of linear equations is a linear representation of relationships between the variables. The goal is to find the point or points where all these linear relationships intersect, which are solutions to the system.
In the given exercise, the equations \(6x - 7y + z = 2\), \(-x - y + 3z = 4\), and \(2x + y - z = 1\) are all linear. Ideally, solving these would identify points in three-dimensional space where these planes intersect. However, the specific ordered triple [in question] did not satisfy any of the equations.