Question

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{r} 2 x-2 y+3 z=6 \\ 4 x-3 y+2 z=0 \\ -2 x+3 y-7 z=1 \end{array}\right. $$

Short answer

The system is inconsistent and has no solution.

Step-by-step solution

- Write down the system of equations

Given the system of equations:a) \(2x - 2y + 3z = 6\) b) \(4x - 3y + 2z = 0\) c) \(-2x + 3y - 7z = 1\)

- Simplify and organize the equations

Take the first and second equations: \(2x - 2y + 3z = 6\) and \(4x - 3y + 2z = 0\).

- Eliminate one variable

Multiply the first equation by 2 to align with the second equation: \(2(2x - 2y + 3z) = 2(6)\) results in \(4x - 4y + 6z = 12\). Now we have: \(4x - 4y + 6z = 12\) and \(4x - 3y + 2z = 0\). Subtract the second from the first: \(4x - 4y + 6z - (4x - 3y + 2z) = 12 - 0\) simplifies to \(-y + 4z = 12\) or \(y = 4z - 12\).

- Substitute in one of the original equations

Substitute \(y = 4z - 12\) into the second original equation: \(4x - 3(4z - 12) + 2z = 0\), which simplifies to \(4x - 12z + 36 + 2z = 0\), thus \(4x - 10z = -36\) or \(2x - 5z = -18\).

- Second elimination step

We now have two equations involving only \(x\) and \(z\): \(2x - 5z = -18\) and the third original \(-2x + 3y - 7z = 1\). Substitute \(y = 4z - 12\) in the third equation: \(-2x + 3(4z - 12) - 7z = 1\), which simplifies to \(-2x + 12z - 36 - 7z = 1\) giving \(-2x + 5z = 37\).

- Solve for one variable

Now we have two new equations: \(2x - 5z = -18\) and \(-2x + 5z = 37\). Add them to eliminate \(x\): \(2x - 5z - 2x + 5z = -18 + 37\) simplifies to \(0 = 19\), which is a contradiction.

- Interpret the result

Since we have encountered a contradiction (0 = 19), the system of equations is inconsistent and has no solution.

Key concepts

inconsistent system

An inconsistent system of equations is one that has no solutions. This means that there are no values for the variables that can simultaneously satisfy all equations in the system. In the given system, after simplifying and manipulating the equations, we encountered the contradictory statement 0 = 19. This contradiction indicates that the system cannot be solved because the equations are mutually exclusive. Such a system is said to be inconsistent because the relationships defined by the equations cannot coexist.

variable elimination

Variable elimination is a method used to solve a system of equations by removing one variable at a time. This simplifies the system and reduces the number of equations to work with. Let's see how this works with our example.
First, we took the equations and aligned them by multiplying the first equation by 2, resulting in two equations with the same coefficient for one of the variables:

• 4x - 4y + 6z = 12
• 4x - 3y + 2z = 0

By subtracting these two equations, we eliminated the variable x and obtained a new equation in terms of y and z. This process continued until we were left with a contradiction, indicating an inconsistent system.

substitution method

The substitution method involves solving one of the equations for one variable and then substituting this expression into the other equations. This method reduces the number of variables in the equations, making them easier to solve.
In our example, we solved for y in terms of z from the equation obtained after eliminating x:

• -y + 4z = 12

From this, we solved for y:

• y = 4z - 12

We then substituted this expression for y into the other equations, gradually reducing the system to one or two variables. This substitution allowed us to eventually detect the inconsistency in the system.

linear equations

Linear equations form the foundation of systems of equations. They are equations that make a straight line when graphed on a coordinate plane, represented by the general form Ax + By + Cz = D.
In our example, we dealt with three linear equations:

• 2x - 2y + 3z = 6
• 4x - 3y + 2z = 0
• -2x + 3y - 7z = 1

Understanding that solutions to these equations would be points (x, y, z) where the planes intersect is crucial. However, as the system was inconsistent, these planes did not have a common intersection point, making it impossible to find such a set of x, y, z that satisfy all three equations.