Problem 10 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the ordered triple given is the solution to the system of equations. $$ \begin{aligned} -x-y+2 z &=3 \\ 5 x+8 y-3 z &=4 \quad \text { and }(4,1,-7) \\ -x+3 y-5 z &=-5 \end{aligned} $$

Step-by-Step Solution

Answer

The ordered triple \((4, 1, -7)\) is not a solution to the system of equations.

Step 1: Substitute in the first equation

The first equation is \(-x-y+2z = 3\). Substitute \((4, 1, -7)\) into this equation:\[-4 - 1 + 2(-7)\]Calculate the expression.

Step 2: Calculate the first equation results

The expression becomes:\[-4 - 1 - 14 = -19\].Since \(-19eq 3\), the ordered triple does not satisfy the first equation.

Step 3: Substitute in the second equation

The second equation is \(5x + 8y - 3z = 4\). Substitute \((4, 1, -7)\) into this equation:\[5(4) + 8(1) - 3(-7)\]Calculate the expression.

Step 4: Calculate the second equation results

Calculate the expression:\[20 + 8 + 21 = 49\].Since \(49eq 4\), the ordered triple does not satisfy the second equation either.

Step 5: Substitute in the third equation

The third equation is \(-x + 3y - 5z = -5\). Substitute \((4, 1, -7)\) into this equation:\[-(4) + 3(1) - 5(-7)\]Calculate the expression.

Step 6: Calculate the third equation results

Calculate the expression:\[-4 + 3 + 35 = 34\].Since \(34eq -5\), the ordered triple does not satisfy the third equation either.

Key Concepts

Ordered Triple

An ordered triple is a set of three numbers, usually represented as \((x, y, z)\). In the context of systems of equations, each number in the ordered triple corresponds to a variable from each equation in the system. The idea is to determine if this specific set of values satisfies all the equations in the system simultaneously.
Understanding ordered triples is crucial because it involves multiple dimensions. Unlike ordered pairs in two-dimensional space (e.g., \((x, y)\)), ordered triples allow us to explore equations in three-dimensional space, which might represent real-world scenarios more comprehensively. For instance, equations involving three variables might represent volume (3D space), with each equation contributing to constraints on the shape or movement of an object.
Finding the correct ordered triple that satisfies all equations in the system indicates that it is a solution to the system. If any equation is not satisfied, then the ordered triple is not a solution to the complete system.

Substitution Method

The substitution method is a popular technique used for finding solutions to systems of equations. It involves expressing one of the variables in terms of the others and substituting this expression into the remaining equations.

• Start by solving one of the equations for a variable of your choice.
• Substitute this expression into the other equations.
• Simplify and solve the resulting equations.

This technique can be particularly useful for systems with three variables, helping to reduce a system's complexity step by step. By substituting, you create a simpler system of equations that is easier to solve. However, be cautious: choosing the wrong variable to substitute might lead to more complicated algebraic expressions. Strategic choice of substitution can significantly streamline the process of finding the ordered triple that satisfies all equations.

Solution Validation

Once you determine a potential solution, in this case, the ordered triple \((4, 1, -7)\), it's essential to validate it by substituting back into the original system of equations. This is the essence of solution validation. You do this by checking each equation:

• Take each variable from the ordered triple.
• Replace the corresponding variable in each equation.
• Simplify and compare the results to the original constants on the right-hand side of each equation.

If all equations hold true, the ordered triple is indeed a valid solution. If even one fails, as in our original exercise where none of the equations were satisfied by \((-19 eq 3)\), \((49 eq 4)\), and \((34 eq -5)\), then the triple is not a solution. This step is crucial to assure the integrity and accuracy of our calculations. Solution validation not only confirms correctness but also helps identify if further adjustments or attempts are necessary.

Algebraic Expressions

Algebraic expressions form the backbone of solving systems of equations. Each equation in our system, such as \(-x - y + 2z = 3\), is an algebraic expression enough to represent some mathematical relationship between variables.
Understanding algebraic expressions involves recognizing:

• The coefficients as multipliers of variables, determining their weight in the equation.
• The constants as the fixed numbers unaffected by the variables' alteration.
• The importance of properly simplifying expressions, especially following substitution.

Algebraic simplification helps consolidate steps and often illuminates the pathway to a solution or identifies inconsistencies. For instance, calculating methods show discrepancies such as \(-4 - 1 - 14 = -19 \) illustrates algebraic expressions can validate or deny potential solutions. They effectively unravel the entities and numbers we interact with, enabling us to solve, manipulate, and understand the structure of equations in a system.