Problem 10

Question

If possible, solve the system. $$ \begin{array}{rr} x-y+z= & -2 \\ x-2 y+z= & 0 \\ y-z= & 1 \end{array} $$

Step-by-Step Solution

Verified

Answer

The solution is $x = -1$, $y = -2$, $z = -3$.

1Step 1: Write the System of Equations

The system of equations is as follows:\[ \begin{align*} x - y + z &= -2 \ x - 2y + z &= 0 \ y - z &= 1 \end{align*} \] We need to solve this system for the variables $x$, $y$, and $z$.

2Step 2: Solve Equation 3 for One Variable

Start with the third equation: $y - z = 1$. Solve for $y$:\[ y = z + 1 \] We will use this expression to substitute $y$ in the other equations.

3Step 3: Substitute in the First Equation

Substitute $y = z + 1$ into the first equation:\[ x - (z + 1) + z = -2 \]This simplifies to:\[ x - 1 = -2 \]Thus, solving for $x$:\[ x = -1 \]

4Step 4: Substitute in the Second Equation

Substitute $y = z + 1$ and $x = -1$ into the second equation:\[ (-1) - 2(z + 1) + z = 0 \]Simplify this expression:\[ -1 - 2z - 2 + z = 0 \]\[ -3 - z = 0 \]Solve for $z$:\[ z = -3 \]

5Step 5: Find All Variables

Now we have $z = -3$ and $x = -1$. Substitute $z = -3$ back into the expression for $y$:\[ y = z + 1 = -3 + 1 = -2 \] Thus, the values are $x = -1$, $y = -2$, and $z = -3$.

6Step 6: Verify the Solution

Substitute $x = -1$, $y = -2$, and $z = -3$ back into the original equations to verify:1. $-1 - (-2) - 3 = -2$ is correct.2. $-1 - 2(-2) - 3 = 0$ is correct.3. $-2 - (-3) = 1$ is correct.The solution satisfies all equations.

Key Concepts

Linear AlgebraMathematical ModelingSolution Verification

Linear Algebra

Linear algebra is a branch of mathematics dealing with vectors, matrices, and systems of linear equations. In essence, linear algebra provides tools to model and solve equations that can be expressed as straight lines in a high-dimensional space.
Systems of linear equations can be represented by a set of equations with the same variables. In the original exercise, we encountered three equations with three unknowns: $x$, $y$, and $z$.

The aim is to find values for these unknowns where all given equations are satisfied simultaneously.
The process typically involves manipulation of the equations to express one variable in terms of another, often making it a "step-by-step" practical approach.

Solutions to these systems can be found using various methods like substitution, elimination, or matrix operations. For our system:$\begin{align*}x - y + z &= -2 \x - 2y + z &= 0 \y - z &= 1\end{align*}$, substitution was used effectively.

Mathematical Modeling

Mathematical modeling with systems of equations allows us to represent real-world scenarios precisely. Each equation in a model often reflects a real-life constraint or a relationship among different factors. The system thereby becomes a structured way to determine unknown variables that are of interest. Imagine we're tasked with understanding certain financial transactions or traffic patterns; these can often be defined by systems of equations.
In the solved system:

The first equation $x - y + z = -2$ might represent a balance of resources, adjustments, or transactional states.
The second, $x - 2y + z = 0$, can indicate a further constraint, additional cost, or regulations.
The third $y - z = 1$ highlights a direct relationship, perhaps preference rules or priorities.

By depicting variables as multi-faceted components of an entire system, linear equations in mathematical modeling help us to find consistent and realistic solutions for decision-making purposes.

Solution Verification

Solution verification is a critical step in solving systems of equations, ensuring that the derived solution satisfies the original conditions stated by the system accurately. Verification acts as a check that the work was carried out correctly during problem-solving.
It involves:

Substituting the values of the variables back into the original equations.
Checking that the left-hand side of each equation equals the right-hand side.
Ensuring no errors occurred during the process for each equation independently.

In the example provided, we were able to find that:

$x = -1$, $y = -2$, and $z = -3$.
Substituting these back yielded: $-1 - (-2) - 3 = -2$, $-1 - 2(-2) - 3 = 0$, and $-2 - (-3) = 1$.
All equations were verified as true, confirming our solution's accuracy.

Verification ensures confidence in the solution, illustrating the reliability and consistency of the mathematical methods used.

Problem 10

Problem 11

Other exercises in this chapter

Problem 10

Write the system of linear equations that the augmented matrix represents. $$ \left[\begin{array}{rrr|r} 1 & -1 & 3 & 2 \\ -2 & 1 & 1 & -2 \\ -1 & 0 & -2 & 1 \e

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Problem 10

Write a symbolic representation for $f(x, y)$ if the function $f$ computes the following quantity. The product of $x^{2}$ and $y^{2}$

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Problem 11

For the given matrices $A$ and $B$ find each of the following. (a) $\boldsymbol{A}+\boldsymbol{B} \quad$ (b) \(\boldsymbol{B}+\boldsymbol{A} \quad(\boldsy

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Problem 11

Graph the solution set to the inequality. $$ x^{2}+y \leq 2 $$

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