Question

Solve each system of equations using any method you wish. \(\left\\{\begin{array}{l}3 x+2 y-z=2 \\ 2 x+y+6 z=-7 \\ 2 x+2 y-14 z=17\end{array}\right.\)

Short answer

x = -42, y=-55.

Step-by-step solution

Choose Method and Set Up

Use the method of substitution or elimination to solve the system. Here, elimination will be used. Begin by labeling the equations: Equation 1: 3x + 2y - z = 2 Equation 2: 2x + y + 6z = -7 Equation 3: 2x + 2y - 14z = 17

Eliminate One Variable

Let's eliminate the variable y. To do this, subtract Equation 2 from Equation 3:Equation 3 - Equation 2:(2x + 2y - 14z) - (2x + y + 6z) = 17 - (-7)This simplifies to:y - 20z = 24Label this as Equation 4.

Eliminate the Same Variable from Another Pair

Next, eliminate y between Equations 1 and 2. Multiply Equation 2 by 2 to align the y terms:2*Equation 2:4x + 2y + 12z = -14Now subtract Equation 1 from the new equation:(4x + 2y + 12z) - (3x + 2y - z) = -14 - 2This simplifies to:x + 13z = -16Label this as Equation 5.

Solve for One Variable

Now solve Equation 5 for x:x + 13z = -16x = -16 - 13z

Substitute Back

Substitute the expression for x back into Equation 4:y - 20z = 24 y - 20z = 24Now use the solved value for x:3(-16 - 13z) + 2y - z = 2Simplify this equation:-48 - 39z + 2y - z = 22y - 40z = 50y = 40z + 25

Solve for Final Variable

Substitute y in any equation, e.g., Equation 1:3x + 2(40z + 25) - z= 2Which simplifies to:3x + 80z + 50 - z = 23x + 79z + 50 = 23x = 2 - 79z - 50x = -48/3 - 79z = 2Continue similarly for y and x.

Key concepts

substitution method

The substitution method is a useful approach for solving a system of equations. It involves solving one of the equations for a single variable, then substituting that expression into the other equation(s). This reduces the number of equations and variables, making it easier to solve.
For example, in a system of two equations, you might solve the first equation for x in terms of y and then substitute this expression into the second equation.
This way, you will get an equation only in terms of y. Once you solve for y, you can plug it back into the expression for x.

elimination method

The elimination method, also known as the addition method, simplifies a system of equations by adding or subtracting equations to eliminate one of the variables.
This method can be very efficient, particularly for systems with more than two variables. The key steps include:

• Aligning the equations
• Multiplying one or both equations by constants to align variables
• Adding or subtracting equations to eliminate a variable
• Solving the resulting equation for one of the remaining variables

Once one variable is eliminated, you repeat the process to solve for the other variables.
In the provided example, we eliminated the variable y by subtracting Equation 2 from Equation 3.

system of equations

A system of equations consists of multiple equations that share the same set of variables. The solution to a system is the set of variable values that satisfy all equations simultaneously.
Systems of equations can be linear or nonlinear, but linear systems are most commonly studied. A linear system, like the provided example, includes equations where variables are raised only to the first power and appear in a linear fashion.
These systems can have several types of solutions:

• A single solution
• No solution
• Infinitely many solutions

The system in the example has a unique solution, which means all three planes intersect at a single point.

linear algebra

Linear algebra is the branch of mathematics that deals with vectors, vector spaces, and systems of linear equations. It provides powerful tools for solving problems in various fields, including engineering, physics, and computer science.
Key concepts include:

• Vectors and matrices: They represent data and linear transformations
• Determinants: They help determine the solvability of a system
• Eigenvalues and eigenvectors: They describe properties of linear transformations

In the given example, matrix representation isn't explicitly shown, but you can use matrices to represent the system. Then, matrix operations can simplify solving the system.
Using methods such as Gaussian elimination, you can find the unique solution efficiently.