Problem 11 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether each ordered triple is a solution of the system of equations. \(\left\\{\begin{array}{rr}4 x+y-z= & 0 \\ -8 x-6 y+z= & -\frac{7}{4} \\ 3 x-y & =-\frac{9}{4}\end{array}\right.\) (a) (0,1,1) (b) \(\left(-\frac{3}{2}, \frac{5}{4},-\frac{5}{4}\right)\) (c) \(\left(-\frac{1}{2}, \frac{3}{4},-\frac{5}{4}\right)\) (d) \(\left(-\frac{1}{2}, \frac{1}{6},-\frac{3}{4}\right)\)

Step-by-Step Solution

Answer

Through analysis, we find that the given ordered triples \(\left(-\frac{3}{2}, \frac{5}{4},-\frac{5}{4}\right)\) and \(\left(-\frac{1}{2}, \frac{3}{4},-\frac{5}{4}\right)\) are solutions to the system of equations. The ordered triples (0,1,1) and \(\left(-\frac{1}{2}, \frac{1}{6},-\frac{3}{4}\right)\) are not solutions to the system of equations.

Step 1: Substitute the values from ordered triple (a) into the system of equations

For the ordered triple (0,1,1), substitute x=0, y=1, and z=1 into the system of equations and verify if they satisfy all the equations. The equations should read: \n\n\(4(0) + 1 - 1 = 0\), \n\n\(-8(0) - 6(1) + 1 = -7/4\) and \n\n\(3(0) - 1 = -9/4\)

Step 2: Evaluate each expression to confirm the equality

For the equations, if they result in true statements then the ordered triple is a solution, otherwise it's not. We find that the first equation holds true, but the second and the third do not, therefore (0,1,1) is not a solution to the system of equations.

Step 3: Repeat the process for the rest of the triples

Repeat the same procedure for the rest of the triples: \n\n\(\left(-\frac{3}{2}, \frac{5}{4},-\frac{5}{4}\right)\), \n\n\(\left(-\frac{1}{2}, \frac{3}{4},-\frac{5}{4}\right)\), and \n\n\(\left(-\frac{1}{2}, \frac{1}{6},-\frac{3}{4}\right)\).\n\n For each triple, substitute the values into the system of equations, simplify and observe if all equations hold true. If they do, then the ordered triple is a solution.

Key Concepts

Ordered Triple

An ordered triple is an essential concept when dealing with systems of equations involving three variables. It is a set of three numbers, such as \(a, b, c\), which represent potential values for the variables \(x, y, z\) in a system of three equations. These numbers are tested to see if they satisfy all given equations in the system.

For instance, in our original exercise, we are provided with several ordered triples such as (0, 1, 1) and \(-\frac{1}{2}, \frac{1}{6}, -\frac{3}{4}\). The purpose is to check if any of these triples make all equations in the system true when substituted into the spots for \(x, y,\) and \(z\).

It is important to ensure the order of the numbers corresponds to the order of the variables in the equations to check the correctness of the solution.

Solution Verification

Solution verification is a crucial part of solving systems of equations. It ensures that the proposed solutions, or ordered triples, actually satisfy every equation in the system. This step is simply about checking correctness.

To verify a solution, you replace the variables in each equation with the respective values from an ordered triple. For example, if you are given the ordered triple (0, 1, 1), you substitute \(x = 0\), \(y = 1\), and \(z = 1\) into each equation. Then, calculate to verify if each equation is true.

During verification:

• If all the substituted equations lead to true statements, the ordered triple is indeed a solution to the system of equations.
• If even one equation is false, then the ordered triple is not a solution.

This meticulous approach ensures no error slips through, maintaining the integrity of the solution.

Substitution Method

The substitution method is an effective strategy to solve systems of linear equations. It's often used to find values of variables one by one.

Here's how it generally works:

• Solve one of the equations for one variable in terms of others. For example, solve for \(x\) from an equation.
• Substitute the expression found for this variable into the other equations.
• This reduces the system to two equations with two variables.
• Solve the reduced system using substitution until you find values for all the variables.

In our scenario, the substitution is slightly different. We substitute the variables directly with proposed values from the ordered triples to verify the solutions rather than reducing the system. It simplifies solution verification because it calculates and checks all equations instantly.

Linear Equations

Linear equations are the foundation for many algebraic problems and are crucial in systems involving multiple equations like ours. A linear equation is any equation that makes a straight line when graphed.

In algebra, a typical form is \ax + by + cz = d\, where \(a, b, c\) are coefficients, and \(d\) is a constant. For a system of linear equations:

• Solutions are graphically represented as intersection points.
• The ordered triple gives coordinates in 3D space where all planes intersect.
• The main goal is to find common values for \(x, y,\) and \(z\) that satisfy all equations, indicating where planes meet.

Understanding linear equations and their graphical interpretations gives deeper insights into solving such systems, since each equation represents a plane in \mathbb{R}^3\.