\left\{\begin{array}{l}x-2 y-z=-1 \\ 2 x+3 y+2 z=-4 \\ 5 x+y-2 z=5\end{array}\right.

The solution to the system of equations is x = − 1 , y = 2 , z = − 4 . This was found by eliminating variables in a step-by-step manner. Verification confirmed that these values satisfy all given equations.
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Answered by Anonymous | 2025-07-04

Eliminate z from equations (2) and (3) to get equation (4): 7 x + 4 y = 1 .
Eliminate z from equations (1) and (2) to get equation (5): 4 x − y = − 6 .
Solve for y in equation (5): y = 4 x + 6 .
Substitute y into equation (4) to find x = − 1 , then find y = 2 , and finally find z = − 4 . The solution is x = − 1 , y = 2 , z = − 4 ​ .

Explanation

Understanding the Problem We are given a system of three linear equations with three unknowns: x, y, and z.

Listing the Equations The equations are:



x − 2 y − z = − 1
2 x + 3 y + 2 z = − 4
5 x + y − 2 z = 5


Stating the Objective Objective: Solve the system of linear equations to find the values of x, y, and z.

Solving the System of Equations Solution plan:

Use the method of elimination or substitution to solve the system of equations.

Step 1: Eliminate z from equations (2) and (3) by adding them: ( 2 x + 3 y + 2 z ) + ( 5 x + y − 2 z ) = − 4 + 5 , which simplifies to 7 x + 4 y = 1 . Call this equation (4).

Step 2: Eliminate z from equations (1) and (2). Multiply equation (1) by 2: 2 ( x − 2 y − z ) = 2 ( − 1 ) , which gives 2 x − 4 y − 2 z = − 2 . Call this equation (1'). Add equation (1') to equation (2): ( 2 x − 4 y − 2 z ) + ( 2 x + 3 y + 2 z ) = − 2 + ( − 4 ) , which simplifies to 4 x − y = − 6 . Call this equation (5).

Step 3: Now we have a system of two equations with two unknowns, x and y: 7 x + 4 y = 1 and 4 x − y = − 6 .

Step 4: Solve for y in equation (5): y = 4 x + 6 .

Step 5: Substitute y = 4 x + 6 into equation (4): 7 x + 4 ( 4 x + 6 ) = 1 , which simplifies to 7 x + 16 x + 24 = 1 , so 23 x = − 23 , and x = − 1 .

Step 6: Substitute x = − 1 into y = 4 x + 6 : y = 4 ( − 1 ) + 6 = 2 .

Step 7: Substitute x = − 1 and y = 2 into equation (1): ( − 1 ) − 2 ( 2 ) − z = − 1 , which simplifies to − 1 − 4 − z = − 1 , so − 5 − z = − 1 , and z = − 4 .

Step 8: The solution is x = − 1 , y = 2 , and z = − 4 .

Verifying the Solution Let's verify the solution by substituting the values of x, y, and z into the original equations: Equation (1): ( − 1 ) − 2 ( 2 ) − ( − 4 ) = − 1 − 4 + 4 = − 1 (Correct) Equation (2): 2 ( − 1 ) + 3 ( 2 ) + 2 ( − 4 ) = − 2 + 6 − 8 = − 4 (Correct) Equation (3): 5 ( − 1 ) + ( 2 ) − 2 ( − 4 ) = − 5 + 2 + 8 = 5 (Correct)

Final Answer The solution to the system of equations is x = − 1 , y = 2 , and z = − 4 .


Examples
Systems of linear equations are used in various fields such as engineering, economics, and computer science. For example, in structural engineering, they can be used to analyze the forces acting on a bridge. In economics, they can be used to model the supply and demand of goods. In computer graphics, they can be used to perform transformations on objects.

Answered by GinnyAnswer | 2025-07-04