How To Solve For 3 Unknowns With 3 Equations

How to Solve for 3 Unknowns with 3 Equations: A Comprehensive Guide

Solving systems of equations is a fundamental concept in algebra with wide-ranging applications in various fields, from physics and engineering to economics and computer science. While solving for one or two unknowns is relatively straightforward, tackling systems with three or more unknowns requires a systematic approach. This comprehensive guide will equip you with the knowledge and techniques to confidently solve systems of three equations with three unknowns. We'll explore three primary methods: elimination, substitution, and using matrices.

Understanding Systems of Equations

Before diving into the solution methods, let's establish a solid understanding of what we're dealing with. A system of three equations with three unknowns is a set of three equations, each containing three variables (typically represented as x, y, and z). The goal is to find the values of x, y, and z that simultaneously satisfy all three equations. These values represent the point of intersection (if one exists) of the three planes defined by the equations in three-dimensional space.

Example:

Consider the following system:

• x + y + z = 6
• 2x - y + z = 3
• x + 2y - z = 3

Our objective is to find the unique values of x, y, and z that satisfy all three equations simultaneously.

Method 1: Elimination

The elimination method involves strategically adding or subtracting equations to eliminate one variable at a time. This process reduces the system to a simpler form that can be easily solved.

Steps:

1. Choose two equations and eliminate one variable: Select any two equations from the system. Multiply one or both equations by constants to make the coefficients of one variable opposites. Then, add the two equations together. This will eliminate that variable, leaving you with an equation in two variables.

2. Repeat the process: Choose a different pair of equations (which may include one from step 1) and eliminate the same variable as in step 1. This will give you another equation in the remaining two variables.

3. Solve the resulting system of two equations: Now you have a system of two equations with two unknowns. Use either elimination or substitution (explained in the next section) to solve for these two variables.

4. Substitute back: Substitute the values obtained in step 3 into any of the original three equations to solve for the third variable.

5. Check your solution: Substitute the values of x, y, and z back into all three original equations to verify that they satisfy all equations simultaneously.

Applying Elimination to our Example:

Let's use the elimination method to solve our example system:

• x + y + z = 6 (Equation 1)
• 2x - y + z = 3 (Equation 2)
• x + 2y - z = 3 (Equation 3)

1. Eliminate z: Add Equation 1 and Equation 3: (x + y + z) + (x + 2y - z) = 6 + 3 => 2x + 3y = 9 (Equation 4)

2. Eliminate z (again): Add Equation 2 and Equation 3: (2x - y + z) + (x + 2y - z) = 3 + 3 => 3x + y = 6 (Equation 5)

3. Solve for x and y: Now we have a system of two equations with two unknowns (Equations 4 and 5):

   • 2x + 3y = 9
   • 3x + y = 6

   We can solve this using elimination again. Multiply Equation 5 by -3: -9x - 3y = -18. Add this to Equation 4: (2x + 3y) + (-9x - 3y) = 9 + (-18) => -7x = -9 => x = 9/7

   Substitute x = 9/7 into Equation 5: 3(9/7) + y = 6 => y = 6 - 27/7 = 15/7

4. Substitute to find z: Substitute x = 9/7 and y = 15/7 into Equation 1: (9/7) + (15/7) + z = 6 => z = 6 - 24/7 = 18/7

5. Check the solution: Substitute x = 9/7, y = 15/7, and z = 18/7 into all three original equations to verify the solution.

Method 2: Substitution

The substitution method involves solving one equation for one variable in terms of the other two variables, and then substituting this expression into the other two equations. This reduces the system to two equations with two unknowns, which can then be solved using substitution or elimination.

Steps:

1. Solve for one variable: Solve one of the equations for one variable in terms of the other two variables.

2. Substitute: Substitute the expression obtained in step 1 into the other two equations. This will give you a system of two equations with two unknowns.

3. Solve the system: Solve the system of two equations using either substitution or elimination.

4. Substitute back: Substitute the values obtained in step 3 back into the expression from step 1 to find the value of the third variable.

5. Check your solution: Verify the solution by substituting the values of x, y, and z into all three original equations.

Applying Substitution to our Example:

Let's apply the substitution method to our example system:

1. Solve for z in Equation 1: z = 6 - x - y

2. Substitute into Equations 2 and 3:

   • 2x - y + (6 - x - y) = 3 => x - 2y = -3
   • x + 2y - (6 - x - y) = 3 => 2x + 3y = 9

3. Solve for x and y: Now we have a system of two equations with two unknowns:

   • x - 2y = -3
   • 2x + 3y = 9

   Solving this system (using either elimination or substitution) yields x = 9/7 and y = 15/7.

4. Substitute to find z: Substitute x = 9/7 and y = 15/7 into z = 6 - x - y: z = 6 - (9/7) - (15/7) = 18/7

5. Check the solution: Verify the solution by substituting x = 9/7, y = 15/7, and z = 18/7 into all three original equations.

Method 3: Using Matrices

The matrix method provides a powerful and efficient way to solve systems of linear equations, particularly for larger systems. This method involves representing the system of equations as an augmented matrix and then performing row operations to obtain the reduced row echelon form.

Steps:

1. Form the augmented matrix: Represent the system of equations as an augmented matrix, where the coefficients of the variables form the coefficient matrix, the constants form the constant matrix, and the two are combined.

2. Perform row operations: Use elementary row operations (swapping rows, multiplying a row by a non-zero constant, adding a multiple of one row to another) to transform the augmented matrix into reduced row echelon form.

3. Extract the solution: The reduced row echelon form will directly provide the values of x, y, and z.

4. Check your solution: Verify the solution by substituting the values of x, y, and z into all three original equations.

Applying Matrices to our Example: The detailed explanation of Gaussian elimination and row reduction using matrices is beyond the scope of a concise blog post but involves a systematic process of manipulating the matrix to achieve the reduced row echelon form. Software like MATLAB, Python's NumPy, or even online matrix calculators can significantly simplify this process. The solution obtained will again be x = 9/7, y = 15/7, and z = 18/7.

Inconsistent and Dependent Systems

It's crucial to understand that not all systems of three equations with three unknowns have a unique solution. Some systems may be inconsistent (having no solution) or dependent (having infinitely many solutions).

• Inconsistent Systems: These systems represent planes that do not intersect at a common point. In the elimination or substitution methods, you'll encounter a contradiction (e.g., 0 = 1). In the matrix method, you'll get a row of zeros with a non-zero constant in the augmented matrix.

• Dependent Systems: These systems represent planes that intersect along a line or coincide. In the elimination or substitution methods, you'll find that one equation is a multiple of another. In the matrix method, you'll have fewer than three leading 1s in the reduced row echelon form.

Conclusion

Solving systems of three equations with three unknowns is a valuable skill with numerous applications. The elimination, substitution, and matrix methods provide powerful tools for finding solutions, when they exist. Remember to always check your solutions by substituting them back into the original equations to ensure accuracy. Understanding the concepts of inconsistent and dependent systems is also crucial for complete mastery of this topic. Practice is key to developing proficiency in solving these types of problems. With consistent effort and the right approach, you'll confidently tackle any system of three equations with three unknowns.