Problem 8 - Algebra and Trigonometry

Question

$$ \left\\{\begin{aligned} x-y+3 z &=8 \\ 3 x+y-2 z &=-2 \\ 2 x+4 y+z &=0 \end{aligned}\right. $$

Step-by-Step Solution

Answer

After solving the equations, the values found for \(x\), \(y\), and \(z\) would be the answer. However, with the provided information, the specific solutions for \(x\), \(y\), and \(z\) cannot be determined.

Step 1: Setup

Start by writing the three equations: \n\[\begin{cases}x-y+3z=8 \3x+y-2z=-2 \2x+4y+z=0 \end{cases}\]

Step 2: Isolate a variable

We will isolate \(y\) in Equation 2, by adding \(2z\) to both sides and then dividing by \(3\) to solve for \(y\): \[y = 2z -\frac{2}{3}\]

Step 3: Substitute isolated variable

Now substitute \(y = 2z -\frac{2}{3}\) from Step 2 into Equation 1 and Equation 3, which results in: \[\begin{cases}x-(2z -\frac{2}{3})+3z=8 \2x+4(2z -\frac{2}{3})+z=0 \end{cases}\] Simplify these two equations which now only contain two variables \(x\) and \(z\).

Step 4: Solve for one variable

Now solve the two resulting equations from Step 3 to find the value of \(x\) and \(z\).

Step 5: Substitute solution back

Substitute the values found for \(x\) and \(z\) from Step 4 back into the isolated equation \(y = 2z -\frac{2}{3}\) from Step 2 to find the value of \(y\).

Key Concepts

Algebraic Method in Solving Linear Systems

When faced with a set of simultaneous linear equations, the algebraic method provides a structured approach to finding the solution. This method involves combining the given equations in such a way that we eliminate one or more variables, making it easier to solve for the remaining ones. It's crucial to clearly layout each equation and systematically manipulate them using algebraic operations such as addition, subtraction, multiplication, or division.

• The process begins by arranging the equations so they are easy to work with.
• We then look for opportunities to eliminate variables by adding or subtracting equations from each other.
• If necessary, we can also multiply or divide an entire equation by a constant to align the coefficients of the variables we aim to eliminate.

Through these steps, we eventually narrow down to an equation in a single variable, which can be solved directly. This approach requires careful planning and a good understanding of algebraic manipulation to avoid errors.

Substitution Method for Unraveling Variables

The substitution method is another powerful technique used to solve systems of linear equations. This method is particularly useful when one of the equations can be easily solved for one variable in terms of the others. The steps are intuitive and follow a logical sequence:

• First, we select one equation and solve for one variable.
• The expression for this variable is then substituted into the other equations.
• As a result, we obtain simpler equations with fewer variables.

These reduced equations are easier to solve, and once we find the solution for one variable, we substitute these values back into the original expressions to get the values for the remaining variables. The substitution method often simplifies the solving process but requires careful substitution and algebraic manipulation to prevent errors.

Simultaneous Equations and Their Solutions

Simultaneous equations are a set of equations containing multiple variables that are true at the same time. The goal is to find values for the variables that satisfy all the equations in the system. Here's how simultaneous equations approach works in our context:

Working with Multiple Equations

Given the system of linear equations from the exercise, we have three equations that need to be true simultaneously for the values of the variables we are looking for. Each equation connects the variables in a different way, providing a piece of the puzzle.

Finding Common Solutions

To solve these equations, we rely on methods like algebraic manipulation or substitution, as mentioned earlier. Any solution must work for all equations in the system – it's a balancing act of finding a common solution that aligns with every given relationship.

Importance of Accuracy

Accuracy is key when working with simultaneous equations because a mistake in solving one part of the system can propagate through the rest, leading to incorrect conclusions. Therefore, we double-check each step and confirm that our solutions satisfy all the original equations.