# Mason's question via Facebook about solving a system of equations (2)

• MHB
• Prove It
In summary, the given system of equations can be solved by using Gaussian elimination to eliminate the z values and then solving for x, y, and z. The final solution is (1, -2, 3).
Prove It
Gold Member
MHB
Solve the following system for \displaystyle \begin{align*} x, y, z \end{align*}:

\displaystyle \begin{align*} z &= 12 - x + 4\,y \\ z &= 4 + 5\,x + 3\,y \\ z &= 5 - 12\,x - 5\,y \end{align*}

As all the z coefficients are the same, it's a good idea to eliminate the z values in the second and third equations, so apply R2 - R1 to R2 and R3 - R1 to R3...

\displaystyle \begin{align*} z &= 12 - x + 4\,y \\ 0 &= -8 + 6\,x - y \\ 0 &= -7 - 11\,x - 9\,y \end{align*}

Now we can multiply the second equation by 9 in order to eliminate the y terms...

\displaystyle \begin{align*} z &= 12 - x + 4\,y \\ 0 &= -72 + 54\,x - 9\,y \\ 0 &= -7 - 11\,x - 9\,y \end{align*}

Now apply R3 - R2 to R3

\displaystyle \begin{align*} z &= 12 - x + 4\,y \\ 0 &= -72 + 54\,x - 9\,y \\ 0 &= 65 - 65\,x \end{align*}

Thus \displaystyle \begin{align*} 65\,x = 65 \implies x = 1 \end{align*}, giving

\displaystyle \begin{align*} 54 \, \left( 1 \right) - 9\,y &= 72 \\ -9\,y &= 18 \\ y &= -2 \end{align*}

and

\displaystyle \begin{align*} z &= 12 - 1 + 4\,\left( -2 \right) \\ z &= 3 \end{align*}

Thus the solution is \displaystyle \begin{align*} \left( x , y, z \right) = \left( 1, -2, 3 \right) \end{align*}.

Equivalently, since z is equal to each of 12−x+4y, −8+6x−y, and −7−11x−9y, they are all equal to each other:
12- x+ 4y= -8+ 6x- y and
-8+ 6x- y= -7- 11x- 9y.

From there, do the same as Prove It.

## 1. How do you solve a system of equations with two variables?

To solve a system of equations with two variables, you can use methods such as substitution, elimination, or graphing. These methods involve manipulating the equations to eliminate one variable and solve for the other.

## 2. Can you provide an example of solving a system of equations with two variables?

Sure, for example, if we have the equations 2x + y = 10 and x - y = 2, we can use the substitution method by solving for one variable in one equation and plugging it into the other. In this case, we can solve for y in the second equation as y = x - 2. Then, we substitute this value into the first equation to get 2x + (x - 2) = 10. Simplifying, we get 3x = 12, and solving for x gives us x = 4. Plugging this value back into the second equation, we get y = 2, so the solution to the system is (4,2).

## 3. What is the importance of solving systems of equations?

Solving systems of equations is important in many fields of science and mathematics, as it allows us to find the values of unknown variables and make predictions or solve real-world problems. It is also a fundamental concept in linear algebra and is often used in optimizing and modeling systems.

## 4. Is there a specific order or method to solving a system of equations?

No, there is no specific order or method that is always used to solve a system of equations. It often depends on the equations given and the preference of the solver. However, some methods may be more efficient or easier to use for certain types of equations.

## 5. What should I do if I get a solution that doesn't make sense when solving a system of equations?

If you encounter a solution that doesn't make sense, such as a negative number for a measurement that cannot be negative, it is likely that there is no solution to the system. This could mean that the equations are inconsistent or contradictory, and there is no way to satisfy all of them at the same time. In this case, it is important to double-check your work and make sure there are no errors in solving the equations.

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