What Is The Solution To The System Of Equations?$\[ \begin{aligned} -x + 2y + Z &= 10 \\ z &= 6 \\ 3x - 2y + 2z &= 8 \end{aligned} \\]A. \[$(0, 2, -6)\$\]B. \[$(-1, 2, 6)\$\]C. \[$(0, 2, 6)\$\]D. \[$(0, -2,

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Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system. In this article, we will explore a system of three linear equations with three variables and find the solution to the system.

The System of Equations

The system of equations we will be working with is:

−x+2y+z=10z=63x−2y+2z=8\begin{aligned} -x + 2y + z &= 10 \\ z &= 6 \\ 3x - 2y + 2z &= 8 \end{aligned}

Step 1: Write Down the Given Equations

The first step in solving a system of linear equations is to write down the given equations. In this case, we have three equations:

  1. −x+2y+z=10-x + 2y + z = 10
  2. z=6z = 6
  3. 3x−2y+2z=83x - 2y + 2z = 8

Step 2: Use the Second Equation to Substitute for z

The second equation tells us that z=6z = 6. We can substitute this value of zz into the first and third equations to get:

  1. −x+2y+6=10-x + 2y + 6 = 10
  2. 3x−2y+2(6)=83x - 2y + 2(6) = 8

Step 3: Simplify the First Equation

We can simplify the first equation by combining like terms:

−x+2y+6=10-x + 2y + 6 = 10

Subtracting 6 from both sides gives us:

−x+2y=4-x + 2y = 4

Step 4: Simplify the Third Equation

We can simplify the third equation by combining like terms:

3x−2y+2(6)=83x - 2y + 2(6) = 8

Expanding the 2(6) gives us:

3x−2y+12=83x - 2y + 12 = 8

Subtracting 12 from both sides gives us:

3x−2y=−43x - 2y = -4

Step 5: Solve the System of Equations

We now have a system of two linear equations with two variables:

  1. −x+2y=4-x + 2y = 4
  2. 3x−2y=−43x - 2y = -4

We can solve this system of equations using either the substitution method or the elimination method. Let's use the elimination method.

Step 6: Multiply the Two Equations by Necessary Multiples

To eliminate the variable yy, we need to multiply the two equations by necessary multiples such that the coefficients of yy in both equations are the same:

  1. Multiply the first equation by 1 and the second equation by 1.

Step 7: Add the Two Equations

Now that the coefficients of yy are the same, we can add the two equations to eliminate the variable yy:

(−x+2y)+(3x−2y)=4+(−4)(-x + 2y) + (3x - 2y) = 4 + (-4)

Simplifying the equation gives us:

2x=02x = 0

Step 8: Solve for x

Dividing both sides of the equation by 2 gives us:

x=0x = 0

Step 9: Substitute x into One of the Original Equations

Now that we have found the value of xx, we can substitute it into one of the original equations to find the value of yy. Let's use the first equation:

−x+2y=4-x + 2y = 4

Substituting x=0x = 0 gives us:

−0+2y=4-0 + 2y = 4

Simplifying the equation gives us:

2y=42y = 4

Dividing both sides of the equation by 2 gives us:

y=2y = 2

Step 10: Find the Value of z

We already know that z=6z = 6 from the second equation.

Conclusion

In conclusion, the solution to the system of equations is:

(x,y,z)=(0,2,6)(x, y, z) = (0, 2, 6)

Therefore, the correct answer is:

C. (0,2,6)(0, 2, 6)

Discussion

This problem is a classic example of a system of linear equations with three variables. We used the substitution method to find the value of zz and then used the elimination method to find the values of xx and yy. The solution to the system of equations is a point in three-dimensional space, and we can visualize it using a graph.

Tips and Tricks

When solving a system of linear equations, it's essential to:

  • Write down the given equations carefully
  • Use the substitution method to find the value of one variable
  • Use the elimination method to find the values of the remaining variables
  • Check your work by plugging the values back into the original equations

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system.

Q: How do I know if a system of linear equations has a solution?

A: A system of linear equations has a solution if and only if the equations are consistent. In other words, if the equations are not contradictory, then the system has a solution.

Q: What are the different methods for solving systems of linear equations?

A: There are several methods for solving systems of linear equations, including:

  • Substitution method
  • Elimination method
  • Graphical method
  • Matrix method

Q: What is the substitution method?

A: The substitution method is a method for solving systems of linear equations where one equation is solved for one variable, and then the value of that variable is substituted into the other equation.

Q: What is the elimination method?

A: The elimination method is a method for solving systems of linear equations where the equations are added or subtracted to eliminate one variable.

Q: How do I choose which method to use?

A: The choice of method depends on the specific system of equations and the variables involved. In general, the substitution method is used when one equation is easily solved for one variable, and the elimination method is used when the equations are easily added or subtracted.

Q: What is the graphical method?

A: The graphical method is a method for solving systems of linear equations where the equations are graphed on a coordinate plane, and the point of intersection is found.

Q: What is the matrix method?

A: The matrix method is a method for solving systems of linear equations where the equations are represented as matrices, and the solution is found using matrix operations.

Q: How do I check my work when solving a system of linear equations?

A: To check your work, plug the values back into the original equations and make sure they are true. If the values satisfy all the equations, then the solution is correct.

Q: What are some common mistakes to avoid when solving systems of linear equations?

A: Some common mistakes to avoid include:

  • Not checking the work
  • Not using the correct method for the specific system of equations
  • Not following the order of operations
  • Not simplifying the equations

Q: How do I know if a system of linear equations has infinitely many solutions?

A: A system of linear equations has infinitely many solutions if the equations are dependent, meaning that one equation is a multiple of the other.

Q: How do I know if a system of linear equations has no solution?

A: A system of linear equations has no solution if the equations are inconsistent, meaning that they are contradictory.

Conclusion

Solving systems of linear equations is an essential skill in mathematics and is used in a wide range of applications, including science, engineering, economics, and computer science. By understanding the different methods for solving systems of linear equations and how to choose the correct method, you can solve systems of linear equations with confidence and accuracy.