Problem 12 - College Algebra with Modeling and Visualization

Question

If possible, solve the system. $$ \begin{aligned} x-y+z &=2 \\ 3 x-2 y+z &=-1 \\ x+y &=-3 \end{aligned} $$

Step-by-Step Solution

Answer

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

Step 1: Assign Equations

Identify and label each equation in the system. Let the equations be: \( E_1: x - y + z = 2 \) \( E_2: 3x - 2y + z = -1 \) and \( E_3: x + y = -3 \).

Step 2: Solve for One Variable from E_3

From equation \( E_3 \), solve for \( x \):\[ x = -3 - y \]

Step 3: Substitute x into E_1

Substitute \( x = -3 - y \) into \( E_1 \):\[ (-3-y) - y + z = 2 \]Simplify to find \( z \):\[ -3 - 2y + z = 2 \Rightarrow z = 5 + 2y \]

Step 4: Substitute x and z into E_2

Substitute \( x = -3 - y \) and \( z = 5 + 2y \) into \( E_2 \):\[ 3(-3-y) - 2y + (5 + 2y) = -1 \]Simplify:\[ -9 - 3y - 2y + 5 + 2y = -1 \Rightarrow -4 - 3y = -1 \]

Step 5: Solve for y

Solve \(-4 - 3y = -1\):\[ -3y = 3 \Rightarrow y = -1 \]

Step 6: Find x and z using y

Substitute \( y = -1 \) back into the equations for \( x \) and \( z \):\[ x = -3 - (-1) = -2 \]\[ z = 5 + 2(-1) = 3 \]

Step 7: Verify the Solution

Check the solution \( (x, y, z) = (-2, -1, 3) \) against all original equations:1. \( -2 - (-1) + 3 = 2 \) is true. 2. \( 3(-2) - 2(-1) + 3 = -1 \) simplifies to \( -6 + 2 + 3 = -1 \) which is true.3. \( -2 + (-1) = -3 \) is true.

Key Concepts

Linear Algebra

Linear Algebra is a branch of mathematics that deals with vectors, vector spaces, and linear equations. In this context, we work with systems of linear equations. These are collections of two or more linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all given equations simultaneously.

When solving systems of linear equations like the one provided, the solution process typically involves finding values for multiple variables which satisfy all the given equations at the same time. This is a foundational concept in linear algebra which helps develop techniques for solving more complex mathematical problems.

• The variables in the equations can represent several measurable elements like quantity or distance.
• Systems can have no solution, one solution, or infinitely many solutions.
• Techniques learned here apply to many fields such as economics, engineering, and the sciences.

Substitution Method

The substitution method is a straightforward and powerful approach in algebra for solving systems of linear equations. It involves solving one of the equations for one variable in terms of the others, then substituting this expression into the other equations.

In our example, we began with the system of equations and solved one of them (\( E_3: x + y = -3 \)) for \( x \). We expressed \( x \) as \( x = -3 - y \). This allowed us to replace \( x \) with \( -3-y \) in the remaining equations, simplifying the system step by step.

• This method is particularly advantageous when one of the equations is easily solvable for a variable.
• It's often useful in systems where there is a clear path to isolate a variable.
• One must perform substitutions carefully to maintain equation balance.

Equation Solving

Equation solving is the process of finding values for the variables that make the equation true. In the context of linear systems, we're looking for a common solution that satisfies all the equations simultaneously.

Once equations are rearranged through substitution or other means, solving becomes a step-by-step numerical process. In this system, after substituting \( x \) and \( z \), we ended up with a single equation in one variable (\( y \)): \( -3y = 3 \). By simply dividing both sides by -3, we determined \( y = -1 \).

• Simplifying equations sometimes requires rearranging terms algebraically.
• Operations like addition, subtraction, multiplication, and division are key tools.
• Solve each variable one at a time to simplify the process.

Verification of Solutions

Verification of solutions is an essential final step when solving systems of equations. This step ensures that the values found for each variable indeed satisfy all original equations in the system.

To verify, substitute the solution back into each equation. For the system provided, substituting \((x, y, z) = (-2, -1, 3)\) into all three original equations shows each equality holds. This confirms correctness and completeness.

• Always verify with the original equations, not the altered or simplified ones.
• If any equation fails, re-check the solution steps for errors.
• Verification builds confidence in the accuracy of the solution.