Problem 35 - Precalculus: Mathematical for Calculus

Question

Solve the system of linear equations. $$\left\\{\begin{aligned} 4 x-3 y+z &=-8 \\ -2 x+y-3 z &=-4 \\ x-y+2 z &=3 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution is \( x = -2 \), \( y = 1 \), \( z = 3 \).

Step 1: Write the System of Equations

The system of equations is given as follows:1. \[ 4x - 3y + z = -8 \]2. \[ -2x + y - 3z = -4 \]3. \[ x - y + 2z = 3 \]

Step 2: Express Variables from Equations

Let's solve the third equation for \(x\):\[ x = y - 2z + 3 \]

Step 3: Substitute into Another Equation

Substitute \( x = y - 2z + 3 \) into the second equation:\[ -2(y - 2z + 3) + y - 3z = -4 \].Simplify the equation:\[ -2y + 4z - 6 + y - 3z = -4 \]Combine like terms:\[ -y + z = 2 \] Thus, we get:\[ y = z - 2 \]

Step 4: Substitute Back to Find All Variables

Now insert \( y = z - 2 \) into \( x = y - 2z + 3 \):\[ x = (z - 2) - 2z + 3 \]Simplify:\[ x = -z + 1 \].

Step 5: Solve for Variables by Substitution

Substitute \( y = z - 2 \) and \( x = -z + 1 \) into the first equation:\[ 4(-z + 1) - 3(z - 2) + z = -8 \].Simplify:\[ -4z + 4 - 3z + 6 + z = -8 \]Combine terms:\[ -6z + 10 = -8 \]Solve for \( z \):\[ -6z = -18 \]\[ z = 3 \]

Step 6: Find Remaining Variables

Using \( z = 3 \):\( y = z - 2 = 3 - 2 = 1 \).\( x = -z + 1 = -3 + 1 = -2 \). The solution is \( x = -2 \), \( y = 1 \), \( z = 3 \).

Key Concepts

Solving Linear Equations

When we talk about solving a system of linear equations, we're looking to find the values of the variables that satisfy all the given equations. Consider each equation as a line in a multidimensional space. The solution is the point where these lines intersect. In our example, we have three variables: \( x \), \( y \), and \( z \), and three equations.

In general, when dealing with linear equations, aim to simplify and reduce the number of variables step by step. The goal is to express one variable in terms of others. This process will streamline the entire solution path by making it easier to substitute and solve successive equations.

Once you've found expressions for some of the variables, substitute back to find specific values. Make sure to double-check calculations, as small errors can lead to incorrect results. Overall, solving these equations accurately relies on a careful balance of algebraic manipulation and logical reasoning.

Substitution Method

The substitution method is a common technique used in solving a system of linear equations. It involves taking an equation and expressing one variable in terms of others. You then substitute this expression into other equations. This helps to gradually eliminate variables and simplify the problem.

In the given example, the substitution method starts by solving the third equation for \( x \), giving us \( x = y - 2z + 3 \). This expression of \( x \) can be plugged into another equation to substitute out \( x \) from the system. Doing this reduces the complexity of the system from three variables to two.
After substituting, simplify the resulting equations by combining like terms. This often reveals clear relationships like \( y = z - 2 \).

Lastly, continue the substitution process until each variable has a single constant or simplified expression. This method ensures a systematic approach to solving systems and is particularly useful when dealing with more complex equations.

Step-by-Step Solution

Providing a step-by-step solution allows for a clear understanding of how each part of the system of equations is resolved. This approach is beneficial because it breaks down the solution into manageable chunks. Each step logically follows from the previous one and paves the way for the next.

To solve the system of equations provided, follow these detailed steps:

• Identify which equations can be solved directly for a single variable. This can initially be done with the third equation for \( x \).
• Utilize the expression obtained from one equation in the others, focusing on simplifying each equation progressively.
• Combine like terms after substitution to further eliminate or isolate variables. For instance, after substitution, \( y \) becomes \( z - 2 \).
• With a simplified setup, you can directly solve for the remaining variable - \( z = 3 \) in the example.
• Back substitute solved values to find the values of other variables. For example, substitute \( z = 3 \) to find \( y \) and \( x \).

This method not only provides a clear path to the answer but also enhances your understanding of each process involved in solving linear equations.