Problem 10 - Precalculus with Limits

Question

Checking Solutions In Exercises \(7 - 10\) , determine whether each ordered triple is a solution of the system of equations. $$\left\\{ \begin{aligned} - 4 x - y - 8 z & = - 6 \\ y + z & = 0 \\ 4 x - 7 y & = 6 \end{aligned} \right.$$ $$\begin{array} { l l } { \text { (a) } ( - 2 , - 2,2 ) } & { \text { (b) } \left( - \frac { 33 } { 2 } , - 10,10 \right) } \\ { \text { (c) } \left( \frac { 1 } { 8 } , - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right) } & { \text { (d) } \left( - \frac { 11 } { 2 } , - 4,4 \right) } \end{array}$$

Step-by-Step Solution

Answer

The details of which ordered triples are solutions would be obtained from carrying out the checks as described in the four steps. Without such specific calculations, there isn't a generic short answer.

Step 1: Check ordered triple (a)

Substitute values from ordered triple (a) (-2, -2, 2) into the three equations. If both sides of each equation are equal, this ordered triple is a solution. Otherwise, it is not.

Step 2: Check ordered triple (b)

Repeat the process with ordered triple (b) \((- \frac{33}{2}, -10, 10)\).

Step 3: Check ordered triple (c)

Next, substitute values from ordered triple (c) \((\frac{1}{8}, -\frac{1}{2}, \frac{1}{2})\).

Step 4: Check ordered triple (d)

Finally, check ordered triple (d) \((-\frac{11}{2}, -4, 4)\).

Key Concepts

Ordered Triple

An ordered triple is a set of three numbers, usually represented as \(x, y, z\). Each number corresponds to a variable in a system of equations. In this context, the ordered triple represents a potential solution to a system with three variables and equations.

In systems of linear equations, an ordered triple is tested by substituting the numbers into the equations. If they satisfy all the equations simultaneously, then it is considered a solution. Otherwise, it is not. When you see an ordered triple like \(-2, -2, 2\), it tells you exactly what numbers to substitute for \(x, y, \, and \z\) in each equation.

Solution Verification

Solution verification is a crucial step while working with systems of equations. This involves substituting the values from the ordered triple into each equation and checking for equality on both sides.

To verify, follow these steps:

• Substitute the values from the ordered triple into each equation one by one.
• Calculate both sides of the equation to see if they match.
• If all equations are true with these values, the ordered triple is a solution.
• If any equation is false, the ordered triple is not a solution.

Verification ensures accuracy and confirms whether the given ordered triple satisfies the system.

Linear Equations

Linear equations form the foundation of systems involving ordered triples. These equations are represented in the form \ax + by + cz = d\ where \a, b, \and \c\ are coefficients and \(d\) is a constant.

A linear equation in three variables \(x, y, \, and \z\) describes a plane in three-dimensional space. Multiple equations can intersect at a single point or line, which represents potential solutions (ordered triples) for the system. Understanding how these equations work helps in visualizing how solutions fit within the framework.

Substitution Method

The substitution method is one common technique used to solve systems of equations, and it's particularly useful here for verifying solutions. This method involves solving one of the equations for one variable and substituting that expression into the other equations.

Steps in the substitution method are:

• Solve one equation for a variable in terms of the others.
• Substitute this expression into the remaining equations.
• Solve these new equations step-by-step, simplifying as needed.
• Finally, back-substitute to find the values of all variables.

This method can simplify complex systems and help find accurate solutions efficiently. It's essential for understanding how ordered triples are checked against systems of equations.