Problem 41 - Algebra and Trigonometry

Question

Solve the system of linear equations. $$ \left\\{\begin{array}{l}{2 x+y+3 z=9} \\ {-x \quad-7 z=10} \\ {3 x+2 y-z=4}\end{array}\right. $$

Step-by-Step Solution

Answer

The system of equations has no solution; it is inconsistent.

Step 1: Write the system of equations

Start by clearly writing out the given system of equations:Equation 1: \( 2x + y + 3z = 9 \)Equation 2: \( -x - 7z = 10 \)Equation 3: \( 3x + 2y - z = 4 \)

Step 2: Solve Equation 2 for x

From Equation 2, solve for \(x\):\[ -x - 7z = 10 \]Rearrange to express \(x\) in terms of \(z\):\[ x = -7z - 10 \]

Step 3: Substitute x in Equation 1

Substitute \(x = -7z - 10\) into Equation 1:\[ 2(-7z - 10) + y + 3z = 9 \]Simplifying gives:\[ -14z - 20 + y + 3z = 9 \]Combine like terms:\[ y - 11z = 29 \]This is now Equation 4.

Step 4: Substitute x in Equation 3

Substitute \(x = -7z - 10\) into Equation 3:\[ 3(-7z - 10) + 2y - z = 4 \]Simplifying gives:\[ -21z - 30 + 2y - z = 4 \]Combine like terms:\[ 2y - 22z = 34 \]Divide the entire equation by 2 to simplify:\[ y - 11z = 17 \]This is now Equation 5.

Step 5: Solve Equation 4 and Equation 5 for y and z

Notice Equations 4 and 5:Equation 4: \( y - 11z = 29 \)Equation 5: \( y - 11z = 17 \)Clearly, there's no solution for consistent \(y\) and \(z\) because 29 does not equal 17, indicating these equations are contradictory.

Step 6: Conclusion

Since Equations 4 and 5 are contradictory, the system of linear equations has no solution; it is inconsistent.

Key Concepts

Inconsistent System

An inconsistent system of linear equations is one that has no possible solutions. This occurs when there's a contradiction among the equations. In simpler terms, it's like having two conflicting demands that can't be true at the same time. Imagine trying to find a single point that satisfies two or more lines, but the lines never intersect.
Take the example from the exercise: After substitutions, we arrived at two new equations:

• Equation 4: \( y - 11z = 29 \)
• Equation 5: \( y - 11z = 17 \)

These are clearly contradictory statements of \( y = 29 + 11z \) and \( y = 17 + 11z \). Hence, there are no values of \( y \) and \( z \) that can satisfy both equations simultaneously. These equations suggest parallel lines with no intersection, confirming the system is indeed inconsistent. As a result, it signifies a set of conditions that cannot coexist in reality.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations and their representations through matrices and vector spaces. It's all about understanding systems of equations, often involving three or more variables. Solving these equations can provide insights into numerous fields, including engineering, physics, and computer science.
In our exercise, we work with a system defined by three linear equations with three variables \( x, y, \) and \( z \). The method of handling these equations often involves techniques like substitution, which we used in the step-by-step solution. By rewriting one variable in terms of others, you can simplify the system greatly. By substituting these expressions back into other equations, you're employing classic linear algebra strategies to find solutions effectively.
Linear algebra allows for such maneuvers, helping us understand multidimensional spaces by breaking down problems into simpler, understandable steps.

Solving Equations

Solving equations, particularly in linear algebra, involves various methods to isolate the variables and find their values. In simple terms, you're playing detective to uncover what numbers make each equation true.
Let's dive into the standard procedure, which often involves strategic manipulation of equations:

• Start by isolating one variable in one equation - just like we did in Step 2, solving for \( x \) from one of the equations.
• Substitute this expression into the other equations. This step helps eliminate one of the variables from the system, reducing the number of unknowns as we did to form Equation 4 and Equation 5.
• Analyze the resulting simplified equations to determine solutions or identify inconsistencies.

Other tools such as graphing can visually represent solutions, but algebraic manipulation remains a powerful and precise method. In doing so, you gain the ability to not only find solutions but also recognize when no solution exists, as it was in this case of an inconsistent system.