Solve 4 Variables and 4 Equations: Step-by-Step Approach

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25-10-2024

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Solving systems of equations with four variables can seem daunting, but with a systematic approach, it can become a manageable task. In this guide, we'll walk through the steps necessary to solve a system of four equations with four variables. This method will help you understand the techniques involved in finding solutions, whether using substitution, elimination, or matrix methods.

Understanding the System of Equations

Let’s say you have the following system of equations:

1. ( 2x + 3y - z + 4w = 7 )
2. ( -x + 4y + 5z - 2w = -3 )
3. ( 3x - 2y + z + 6w = 10 )
4. ( x + y - 3z + w = 1 )

In this system, (x), (y), (z), and (w) are the variables we need to solve for.

Step 1: Organize the Equations

Before solving the equations, it’s important to organize them. Align the variables and constants neatly:

Step 2: Choose a Method

You can solve this system using several methods: substitution, elimination, or using matrices. Here we will utilize the elimination method for clarity and simplicity.

Step 3: Eliminate One Variable

Let’s start by eliminating (z) from the equations. We can choose two equations to create a new equation without (z).

From equation 1 and equation 3, we can manipulate them as follows:

• Multiply equation 1 by 1: [ 2x + 3y - z + 4w = 7 ]

• Multiply equation 3 by -1: [ -3x + 2y - z - 6w = -10 ]

Step 4: Add or Subtract Equations

Now, add these two equations:

[ (2x + 3y - z + 4w) + (-3x + 2y - z - 6w) = 7 - 10 ]

This simplifies to:

[ -x + 5y - 2z - 2w = -3 ]

Now we have a new equation without (z).

Step 5: Repeat the Process

Repeat this process with the other equations to eliminate (z) as follows:

From equation 2 and equation 4:

• Equation 2: [ -x + 4y + 5z - 2w = -3 ]

• Equation 4: [ x + y - 3z + w = 1 ]

Multiply equation 4 by (3):

[ 3x + 3y - 9z + 3w = 3 ]

Now we have:

[ -x + 4y + 5z - 2w + (3x + 3y - 9z + 3w) = -3 + 3 ]

This results in:

[ 2x + 7y - 4z + w = 0 ]

Step 6: Build the New System

Now you can build a new system of equations with the remaining variables (x), (y), and (w):

1. (-x + 5y - 2w = -3)
2. (2x + 7y + w = 0)

Step 7: Solve the New System

Now, you can solve the simplified system:

From the first equation, express (w):

[ w = -3 + x - 5y ]

Substituting (w) into the second equation:

[ 2x + 7y + (-3 + x - 5y) = 0 ]

This simplifies to:

[ 3x + 2y - 3 = 0 \implies 3x + 2y = 3 ]

Step 8: Express One Variable in Terms of Another

From (3x + 2y = 3):

[ y = \frac{3 - 3x}{2} ]

Now substitute (y) back into the expression for (w) and you can find (w) in terms of (x):

Step 9: Back-Substitution

With (y) expressed in terms of (x), substitute back into the earlier equations to find the values for (x), (y), (z), and (w).

Final Note

By systematically eliminating variables and substituting back, you can find the values of (x), (y), (z), and (w).

Important Note: Each time you manipulate equations, make sure to check for consistency. If at any point you derive a contradiction (like (0 = 1)), the system does not have a solution.

Using these steps, you can tackle any system of four variables and equations. With practice, solving these equations will become easier, and you'll enhance your problem-solving skills significantly! 💪