Problem 80 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the two systems of linear equations yield the same solution. If so, find the solution. $$\left\\{\begin{aligned} x+3 y-z=& 19 \\ y+6 z=&-18 \\ z=&-4 \end{aligned}\left\\{\begin{array}{rr} x-y+3 z= & -21 \\ y-2 z= & 14 \\ z= & -4 \end{array}\right.\right.$$

Step-by-Step Solution

Answer

Yes, both systems of equations yield the same solution: \((-3,6,-4)\).

Step 1: Solving the First System of Equations

Starting with the first system of equations: \[x+3 y-z=19, y+6 z=-18, z=-4\] Since \(z\) is given as \(-4\), each instance of \(z\) in the first two equations can be replaced with \(-4\). This substitution yields: \[x + 3y +4 =19, y -24=-18\] Leading to: \[x + 3y =15, y =6\] Then, substituting \(y=6\) into the first equation gives: \[x = 15 - 3*6 = -3\]

Step 2: Solving the Second System of Equations

Next, apply similar strategies to the second system of equations: \[x-y+3 z=-21, y-2 z=14, z=-4\] Again, replace \(z\) with \(-4\): \[x - y - 12 = -21, y + 8 = 14\] Simplifying these gives: \[x - y = -9, y = 6\] Finally, substituting \(y=6\) into the first equation yields: \[x = -9 + 6 = -3\]

Step 3: Comparing the Solutions

Comparing the solutions obtained from both systems of equations reveals that \((x,y,z)\) ends up as \((-3,6,-4)\) in both scenarios. Therefore, both systems of equations yield the same solution of \((-3,6,-4)\).

Key Concepts

Solution of Linear Systems

Linear systems consist of multiple linear equations that we often solve to find the values of variables that satisfy all equations simultaneously. Solving these systems can help us understand relationships within data, providing insights into both practical and theoretical problems.
In solving a linear system, the goal is to identify the set of values for the variables that make each equation true at the same time. This system can either have a unique solution, no solution, or infinitely many solutions.

• A unique solution occurs when the system has a single set of values for the variables that satisfy all equations.
• No solution indicates that there is no possible way to find values that satisfy all the equations simultaneously.
• Infinite solutions imply that there are several sets of values that can make the equations true.

Linear systems can often be solved using methods such as substitution, elimination, and graphical representation.

Substitution Method

The substitution method is a common technique for solving linear systems of equations. This approach involves isolating one variable in one of the equations and then substituting the expression for that variable into the other equations. By doing so, we reduce the problem to solving for fewer variables.
Let's break down the substitution process:

• Select one of the equations with a variable that can be easily isolated.
• Solve this equation for the isolated variable to get an expression, for example, if you have an equation like \( x + 3y = 15 \), solve it for \( x \), which gives you \( x = 15 - 3y \).
• Substitute the expression found in the previous step into the other equations. This reduces their number of variables.

After substitution, you will solve for one remaining variable. Once found, you substitute it back to find the other variable values.

Consistent Systems

A consistent system of linear equations is one that has at least one solution. Consistency is a vital concept because it tells us whether our linear system can be solved. Systems are categorized based on the number of solutions they produce:

• A consistent system has either one distinct solution or infinitely many solutions.
• An inconsistent system has no solutions, indicating that the equations contradict each other.

In practical terms, a consistent system implies that there is a common point or several points on the graph where all equations intersect.
Determining consistency is a fundamental step before solving a system, as it saves time and guides us on what to expect in terms of solutions.

Simultaneous Equations

Simultaneous equations are sets of equations that involve the same set of variables. These equations are solved together to find common variable values that satisfy each equation within the set.
Solving simultaneous equations often requires methods like substitution or elimination. The primary goal is to reduce the complexity of the system, making it easier to find solutions:

• Start by simplifying the system as much as possible by combining or manipulating the equations.
• Then, use either the substitution method or elimination method to solve for the variables.

By solving them, we can find values that make each equation true at the same time, giving us a clear set of solutions.