Problem 9 - Algebra and Trigonometry

Question

Use back-substitution to solve the triangular system. $$ \left\\{\begin{aligned} 2 x-y+6 z &=5 \\ y+4 z &=0 \\\\-2 z &=1 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solution is \( x = 5 \), \( y = 2 \), \( z = -\frac{1}{2} \).

Step 1: Solve the last equation for z

We start with the last equation, \-2z = 1\. To isolate \ z\, divide both sides of the equation by \-2\: \[ z = \frac{1}{-2} = -\frac{1}{2} \]

Step 2: Substitute z into the second equation

Substitute \ z = -\frac{1}{2} \ into the second equation \ y + 4z = 0 \: \[ y + 4\left(-\frac{1}{2}\right) = 0 \] \[ y - 2 = 0 \] Solve for \ y \ by adding 2 to both sides: \[ y = 2 \]

Step 3: Substitute y and z into the first equation

Substitute \ y = 2 \ and \ z = -\frac{1}{2} \ into the first equation \ 2x - y + 6z = 5 \: \[ 2x - 2 + 6\left(-\frac{1}{2}\right) = 5 \] Simplifying gives: \[ 2x - 2 - 3 = 5 \] \[ 2x - 5 = 5 \] Add 5 to both sides to isolate \ 2x \: \[ 2x = 10 \] Divide by 2 to solve for \ x \: \[ x = 5 \]

Key Concepts

Triangular System

A triangular system is a specific type of system of equations where the equations are arranged in a way that they naturally lead to a step-by-step solution. In this setup, one of the equations contains only one variable, another equation contains two variables, and another may contain three, and so on. This arrangement is typically represented in a triangular, upper or lower form where each row starts with one more zero than the previous one, simplifying the problem at hand. In the provided exercise, we see a lower triangular system:

• The last equation is \-2z = 1\, which involves only one variable, making it easy to solve directly for \(z\).
• The second equation, \(y + 4z = 0\), contains both \(y\) and \(z\), and is solved after determining \(z\).
• The first equation, \(2x - y + 6z = 5\), involves \(x\), \(y\), and \(z\), solved last using the outcomes from the previous equations.

This method simplifies the process of solving equations sequentially rather than dealing with all the variables at once.

System of Equations

A system of equations is essentially a set of two or more equations that we solve simultaneously to find a common solution. These equations share variables and we need to find values that satisfy all equations in the system at the same time. Systems of equations can be linear or nonlinear, with linear being the simplest form.In a linear system, the equations are all first-degree equations, meaning they graph as straight lines. The solution to a system of linear equations is the point or points where the lines intersect, representing the values that satisfy all equations at once. For our triangular system:

• The system has three equations with three variables \(x, y, \) and \(z\).
• Solving this set requires reducing the complexity by solving one variable at a time and working upwards through the equations.
• The approach used was the back-substitution method, which is effective when dealing with triangular systems.

This strategy leverages the simplicity of dealing with equations that are already simplified to focus on individual variables.

Algebra

Algebra is an essential mathematical language used to describe relationships using symbols, often to solve for unknown values. In solving systems of equations, algebraic manipulation plays a crucial role. We rely on algebraic principles to rearrange and simplify equations, making it possible to isolate variables and find solutions.Key algebraic techniques used in the exercise include:

• Equation rearrangement: Adjusting terms to isolate a variable on one side of the equation.
• Substitution: This is replacing a variable in one equation with the expression derived from another, allowing you to simplify and solve step by step.
• Arithmetic operations: Basic operations like addition, subtraction, multiplication, and division are used to solve expressions after substitution, as seen in solving for \(x, y,\) and \(z\).

Through algebra, even complex systems can be systematically solved, leveraging its logic and structure to break down and manage each part of a problem.