Solve the following $3 \times 3$ system. Enter the coordinates of the solution below.
$\begin{aligned}
2 x-3 y-2 z & =4 \
x+3 y+2 z & =-7 \
-4 x-4 y-2 z & =10
\end{aligned}$

Eliminate z by adding equations 1 and 2, resulting in 3 x = − 3 , so x = − 1 .
Eliminate z by adding equations 2 and 3, resulting in − 3 x − y = 3 .
Substitute x = − 1 into − 3 x − y = 3 to find y = 0 .
Substitute x = − 1 and y = 0 into x + 3 y + 2 z = − 7 to find z = − 3 .
The solution is ( − 1 , 0 , − 3 ) ​ .

Explanation

Analyzing the System of Equations We are given a system of three linear equations in three variables: x , y , and z . Our goal is to find the values of x , y , and z that satisfy all three equations simultaneously. The equations are:

2 x − 3 y − 2 z ​ = 4 x + 3 y + 2 z ​ = − 7 − 4 x − 4 y − 2 z ​ = 10 ​

Eliminating z from Equations 1 and 2 To solve this system, we can use the method of elimination. First, we add the first and second equations to eliminate z :

( 2 x − 3 y − 2 z ) + ( x + 3 y + 2 z ) = 4 + ( − 7 ) 3 x = − 3 x = − 1

Eliminating z from Equations 2 and 3 Next, we add the second and third equations to eliminate z :

( x + 3 y + 2 z ) + ( − 4 x − 4 y − 2 z ) = − 7 + 10 − 3 x − y = 3

Solving for y Now we substitute the value of x = − 1 into the equation − 3 x − y = 3 :

− 3 ( − 1 ) − y = 3 3 − y = 3 − y = 0 y = 0

Solving for z Finally, we substitute the values of x = − 1 and y = 0 into the second equation to solve for z :

x + 3 y + 2 z = − 7 ( − 1 ) + 3 ( 0 ) + 2 z = − 7 − 1 + 2 z = − 7 2 z = − 6 z = − 3

Stating the Solution Therefore, the solution to the system of equations is x = − 1 , y = 0 , and z = − 3 . We can write this as an ordered triple ( − 1 , 0 , − 3 ) .

Examples
Systems of equations are used in various fields such as engineering, economics, and computer science. For example, in electrical engineering, systems of equations can be used to analyze circuits and determine the current and voltage in different parts of the circuit. In economics, they can be used to model supply and demand curves and determine equilibrium prices and quantities. In computer graphics, they can be used to perform transformations on objects in 3D space.

Answered by GinnyAnswer | 2025-07-07

The solution to the system of equations is given by the ordered triple (-1, 0, -3). This was found by strategically using substitution and elimination methods on the equations provided. Each variable was derived step-by-step to ensure accuracy in the solution.
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Answered by Anonymous | 2025-08-25