Solving Systems of 3 Equations in Algebra 2

In Algebra 2, you encounter a new level of complexity when dealing with systems of equations. While solving two equations with two unknowns is relatively straightforward, solving three equations with three unknowns requires a more systematic approach. This blog post will guide you through the process of solving systems of three equations using the elimination method, a powerful technique that simplifies the problem.

Understanding the Concept

Imagine you have three equations with three variables, let's say x, y, and z. Each equation represents a plane in three-dimensional space. The solution to this system of equations is the point where all three planes intersect. Finding this point is the goal of solving the system.

The Elimination Method

The elimination method works by systematically eliminating variables until you are left with a single equation with one unknown. Here's how it works:

1. Choose two equations and eliminate one variable. This is done by multiplying one or both equations by a constant so that the coefficients of the variable you want to eliminate are opposites. Then, add the two equations together. The variable with opposite coefficients will cancel out.
2. Repeat step 1 with a different pair of equations. Choose a different pair of equations from the original system and eliminate the same variable you eliminated in step 1. This will give you a second equation with two unknowns.
3. Solve the system of two equations. You now have two equations with two unknowns. Use any method you prefer (substitution or elimination) to solve for these two variables.
4. Substitute the values you found back into one of the original equations. This will allow you to solve for the remaining variable.

Example

Let's solve the following system of equations:

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

Step 1: Eliminate z from Equation 1 and Equation 2

Multiply Equation 1 by -1:

• -x - y - z = -6

Add this modified equation to Equation 2:

• -x - y - z = -6
• 2x - y + z = 3
• --------------------
• x - 2y = -3 (Equation 4)

Step 2: Eliminate z from Equation 1 and Equation 3

Add Equation 1 and Equation 3:

• x + y + z = 6
• x + 2y - z = 1
• --------------------
• 2x + 3y = 7 (Equation 5)

Step 3: Solve the system of Equation 4 and Equation 5

Multiply Equation 4 by -2:

• -2x + 4y = 6

Add this modified equation to Equation 5:

• -2x + 4y = 6
• 2x + 3y = 7
• --------------------
• 7y = 13
• y = 13/7

Substitute y = 13/7 into Equation 4 to solve for x:

• x - 2(13/7) = -3
• x = -3 + 26/7
• x = 5/7

Step 4: Substitute x and y into Equation 1 to solve for z

• (5/7) + (13/7) + z = 6
• z = 6 - 18/7
• z = 24/7

Therefore, the solution to the system of equations is:

• x = 5/7
• y = 13/7
• z = 24/7

Conclusion

Solving systems of three equations with three unknowns using the elimination method can seem daunting at first, but with practice, it becomes a manageable process. Remember to systematically eliminate variables, carefully track your equations, and double-check your calculations to ensure accuracy.

This method is crucial for understanding concepts in higher-level mathematics, physics, and engineering, where systems of equations are commonly used to model real-world phenomena.