Question

Verify that the values of the variables listed are solutions of the system of equations. $$ \begin{array}{l} \left\\{\begin{array}{l} 4 x-z=7 \\ 8 x+5 y-z=0 \\ -x-y+5 z=6 \\ \end{array}\right.\\\ x=2, y=-3, z=1 \\ (2,-3,1) \end{array} $$

Short answer

The values (2, -3, 1) satisfy all three equations.

Step-by-step solution

- Substitute first equation

Substitute the given values of the variables into the first equation, which is 4x - z = 7: 4(2) - 1 = 8 - 1 = 7. This satisfies the first equation.

- Substitute second equation

Substitute the given values of the variables into the second equation, which is 8x + 5y - z = 0: 8(2) + 5(-3) - 1 = 16 - 15 - 1 = 0. This satisfies the second equation.

- Substitute third equation

Substitute the given values of the variables into the third equation, which is -x - y + 5z = 6: -2 - (-3) + 5(1) = -2 + 3 + 5 = 6. This satisfies the third equation.

Key concepts

Substitution Method

The substitution method is a way of solving systems of equations by substituting one equation into another. This involves solving one of the equations for one variable and then plugging that expression into the other equation. This technique simplifies the system, making it easier to find the values of the variables. In our example, we verified the proposed solution \(x=2, y=-3, z=1\) by directly substituting these values into each equation. This step-by-step method ensures that each solution is valid.

Verification of Solutions

Verification is a critical step in confirming that the provided solutions are accurate. After proposing solutions such as (2, -3, 1), verification involves substituting these values back into the original equations. This process ensures that each equation holds true.
Let's break it down:

• Substitute the values into the first equation: \(4x - z = 7\). When the values are substituted, \(4(2) - 1 = 8 - 1 = 7\), it confirms the solution fits.
• Substitute into the second equation: \(8x + 5y - z = 0\). Plugging the values gives \(8(2) + 5(-3) - 1 = 16 - 15 - 1 = 0\), again confirming the solution.
• Lastly, substitute into the third equation: \(-x - y + 5z = 6\). By substituting, \(-2 - (-3) + 5(1) = -2 + 3 + 5 = 6\) confirms the solution.

This verification confirms that \(x=2, y=-3, z=1\) is indeed the solution to the system.

Linear Equations

A linear equation describes a relationship between two variables and plots a straight line when graphed. In a system of linear equations, we're dealing with multiple such relationships. The goal is to find the point where these lines intersect, representing the solution.
The given system of equations \(4x - z = 7, 8x + 5y - z = 0, -x - y + 5z = 6\) forms a 3D space intersection. Each substitution showed that all equations were satisfied with \(x = 2, y = -3, z = 1\).
These solutions show where the three planes formed by the equations intersect, confirming the unique solution to this system of linear equations.