Problem 38 - Precalculus: Mathematical for Calculus

Question

Solve the system of linear equations. $$\left\\{\begin{array}{l} 3 x-y+2 z=-1 \\ 4 x-2 y+z=-7 \\ -x+3 y-2 z=-1 \end{array}\right.$$

Step-by-Step Solution

Answer

\(x = -2\), \(y = 1\), and \(z = 3\).

Step 1: Write the System of Equations

We start by writing down the given system of linear equations: \[\begin{align*}3x - y + 2z &= -1 \4x - 2y + z &= -7 \-x + 3y - 2z &= -1\end{align*}\]

Step 2: Express One Variable in Terms of Others

Let's solve the first equation for \( y \):\[ y = 3x + 2z + 1 \]

Step 3: Substitute Expression into Other Equations

Substitute \( y = 3x + 2z + 1 \) into the second and third equations:\[\begin{align*}4x - 2(3x + 2z + 1) + z &= -7 \-x + 3(3x + 2z + 1) - 2z &= -1 \end{align*}\]

Step 4: Simplify the Second Equation

Simplify the second equation:\[4x - 6x - 4z - 2 + z = -7 \ -2x - 3z = -5\]

Step 5: Simplify the Third Equation

Simplify the third equation:\[-x + 9x + 6z + 3 - 2z = -1 \ 8x + 4z = -4 \]

Step 6: Solve the Simplified System

We now solve the new system of equations:\[\begin{align*}-2x - 3z &= -5 \ 8x + 4z &= -4 \end{align*}\]Multiply the first equation by 4:\[-8x - 12z = -20\]Add this to the second equation:\[-8x - 12z + 8x + 4z = -20 - 4 \-8z = -24 \z = 3\]

Step 7: Substitute Back to Find x

Substitute \( z = 3 \) back into one of the equations, e.g., \[-2x - 3(3) = -5\]which simplifies to:\[-2x - 9 = -5 \-2x = 4 \x = -2\]

Step 8: Substitute to Find y

Finally, substitute both \( x = -2 \) and \( z = 3 \) into \( y = 3x + 2z + 1 \):\[ y = 3(-2) + 2(3) + 1 \y = -6 + 6 + 1 \y = 1\]

Step 9: Verify the Solution

Verify by substituting \( x = -2 \), \( y = 1 \), \( z = 3 \) into all original equations:\[\begin{align*}3(-2) - 1 + 2(3) &= -1 \4(-2) - 2(1) + 3 &= -7 \-(-2) + 3(1) - 2(3) &= -1\end{align*}\]All hold true, confirming the solution.

Key Concepts

Substitution Method

The substitution method is one of the most effective techniques for solving a system of equations. It involves expressing one variable in terms of the others and substituting this expression into the other equations. The main goal is to reduce the number of variables, making the equations easier to solve. In our example, we started by isolating the variable \( y \) from the first equation:\[ y = 3x + 2z + 1 \]This step converted the problem into two simpler equations. By replacing all occurrences of \( y \) in the other equations with \( 3x + 2z + 1 \), we reduced the system to a manageable form. This method is especially useful in equations with more than two variables and can be applied iteratively if needed.

• It's important to choose the simplest equation to express one variable in terms of the others, this reduces potential errors and leads to a quicker solution.
• The substitution method is versatile and can also solve non-linear systems of equations.

Like any algebraic method, substitution requires careful handling of algebraic manipulation to avoid mistakes.

Linear Algebra

Linear algebra centers around the study of vectors, vector spaces, and systems of linear equations. A system of linear equations can have one solution, infinitely many solutions, or no solution at all. To solve these, we often rely on concepts like row operations, matrices, and vector spaces. In the context of this exercise, understanding how to represent and manipulate equations is crucial. When dealing with a system of equations like ours consisting of three variables and three equations, linear algebra provides tools and methodologies such as substitution and elimination. These methods are part of a broader set of strategies that include matrix operations, which are particularly useful for larger systems.

• Linear algebra helps in transforming complex problems into solvable equations.
• It's foundational for fields like engineering, computer science, and economics, where modeling real-world systems often leads to solving linear equations.

Grasping linear algebra concepts is essential for understanding how the solutions are derived and verified, forming a solid mathematical foundation.

Equation Simplification

Equation simplification involves reducing equations to their simplest form to make them easier to solve. In solving systems of linear equations, simplification is critical for clarity and reducing computation errors. After substituting variables, we often end up with equations that seem complex at first.For instance, upon substituting \( y = 3x + 2z + 1 \) into the second and third equations, the resulting equations required simplification:1. For the second equation: \[ 4x - 2(3x + 2z + 1) + z = -7 \] simplification leads to: \[ -2x - 3z = -5 \]2. For the third equation:\[ -x + 3(3x + 2z + 1) - 2z = -1 \] which simplifies to:\[ 8x + 4z = -4 \]

• Simplification involves combining like terms, factoring, and eliminating unnecessary components.
• It provides a clearer path to finding the values of the unknowns.

Efficient simplification ensures that each step of solving the system is clear and that algebraic errors are minimized. It is an invaluable skill in solving equations accurately and efficiently.