Solving Systems Of Equations: A Step-by-Step Guide

Hey math enthusiasts! Ever stumbled upon a system of equations and felt a little lost? Don't worry, it's a common feeling. But with the right approach, solving these problems can become quite manageable. In this article, we'll dive into how to solve systems of equations, specifically focusing on a set of three equations with three variables. We'll break down the process step-by-step, making it easy to follow along. So, grab your pencils and let's get started!

Understanding the Basics: What are Systems of Equations?

Before we jump into the solution, let's quickly review what a system of equations is all about. A system of equations is simply a set of two or more equations that we aim to solve simultaneously. This means we're looking for values for the variables (usually x, y, and z) that satisfy all equations in the system. When we have three equations, as in the example provided, and three unknowns (x, y, and z), we are typically able to find a unique solution—a single set of values that works for all three equations. However, it's also possible for a system to have no solution (if the equations are inconsistent) or infinitely many solutions (if the equations are dependent).

Think of each equation as a plane in three-dimensional space. The solution to the system is the point (or points) where all the planes intersect. If the planes don't intersect at a single point, then there's no unique solution. If the planes intersect along a line or coincide, then there are infinitely many solutions. This visual understanding can be helpful, but for the purpose of solving, we primarily rely on algebraic methods. We'll be using a combination of techniques, like elimination and substitution, to systematically reduce the system until we isolate each variable.

The system we're tackling here has three equations. This might seem daunting at first, but the principle is the same as with two equations: we aim to eliminate variables until we can solve for one. Then, we back-substitute to find the values of the other variables. The beauty of solving these systems lies in the methodical approach. By sticking to a clear set of steps, we can solve complex problems without getting overwhelmed. So let's get into the practical steps of solving the provided system of equations. Remember, the goal is to find the values of x, y, and z that satisfy all three equations.

Step-by-Step Solution: The Elimination Method

Let's tackle the system of equations you provided:

$$\begin{aligned} 3 x+3 y+z & =-20 \\ x-3 y+2 z & =3 \\ 8 x-2 y+3 z & =-33 \end{aligned}$$

Our goal is to find the values of x, y, and z that satisfy all three equations simultaneously. We'll use the elimination method, a systematic approach to simplify the equations step by step. This method involves manipulating the equations to eliminate one variable at a time until we can easily solve for the others. Here's how we'll proceed:

Step 1: Eliminate y from two pairs of equations.

• Pair 1: Equations 1 and 2. Notice that the y terms have opposite coefficients (+3 and -3). We can simply add the first and second equations to eliminate y:

  (3x + 3y + z) + (x - 3y + 2z) = -20 + 3

  This simplifies to:

  4x + 3z = -17 (Equation 4)

• Pair 2: Equations 2 and 3. To eliminate y here, we'll need to make the y coefficients opposites. Multiply Equation 2 by -2/3 to get 2y and then add it to equation 3.

  (-2/3) * (x - 3y + 2z) = -2/3 * 3

  This simplifies to:

  (-2/3)x + 2y - (4/3)z = -2

  Now, add this result to Equation 3:

  (8x - 2y + 3z) + ((-2/3)x + 2y - (4/3)z) = -33 + (-2)

  This simplifies to:

  (22/3)x + (5/3)z = -35 (Equation 5)

Step 2: Eliminate z from Equations 4 and 5.

• Now we have two equations (Equation 4 and Equation 5) with two variables (x and z). We can eliminate z using a similar approach. Multiply equation 4 by -5/3

  (-5/3)(4x* + 3z) = -5/3*(-17)

  This simplifies to:

  (-20/3)x - 5z = 85/3

  Then add this equation to equation 5:

  ((22/3)x + (5/3)z) + ((-20/3)x - 5z) = -35 + 85/3

  This simplifies to:

  (2/3)x - (10/3)z = -20/3

  Multiply by 3:

  2x - 10z = -20

  Dividing by 2

  x - 5z = -10

  Equation 6

  Multiply Equation 4 by 5/3:

  (5/3)(4x* + 3z) = (5/3)*(-17)

  This simplifies to:

  (20/3)x + 5z = -85/3

  Add this equation to Equation 5:

  ((22/3)x + (5/3)z) + ((20/3)x + 5z) = -35 + (-85/3)

  This simplifies to:

  (42/3)x + (20/3)z = -190/3

  Multiply by 3:

  42x + 20z = -190

  Dividing by 2

  21x + 10z = -95

• Pair 2: Equations 4 and 5. Now we must eliminate z. To do this, multiply Equation 4 by -5/3 and add the result to Equation 5.

  (-5/3)(4x* + 3z) = (-5/3)(-17) => -20/3x* - 5z = 85/3

  Adding this to Equation 5, we have:

  (-20/3x - 5z) + (22/3x + 5/3z) = 85/3 + (-35)

  Which simplifies to:

  2/3x - 10/3z = -20/3

  Then, by multiplying by 3, we get: 2x - 10z = -20. Dividing by 2, we have x - 5z = -10 (Equation 6).

Step 3: Solve for x and z.

• From the equations above, we already have our simplified system:

  4x + 3z = -17

  x - 5z = -10

• Multiply the second equation by -4:

  -4x + 20z = 40

  Add this result to 4x + 3z = -17

  (-4x + 20z) + (4x + 3z) = 40 + (-17)

  Which simplifies to:

  23z = 23

  So, z = 1.

• Substitute z=1 into x - 5z = -10:

  x - 5*(1) = -10

  x = -5

Step 4: Solve for y.

• Now that we have the values of x and z, we can substitute them into any of the original equations to solve for y. Let's use the first equation:

  3x + 3y + z = -20

• Substitute x = -5 and z = 1:

  3*(-5) + 3y + 1 = -20

  -15 + 3y + 1 = -20

  3y = -6

  y = -2

Step 5: State the Solution.

Therefore, the solution to the system of equations is x = -5, y = -2, and z = 1. We can write this as an ordered triple: (-5, -2, 1).

Verification: Checking Your Answer

It's always a good practice to verify your solution. Substitute the values of x, y, and z back into the original equations to ensure they hold true. For example, using the first equation:

3*(-5) + 3*(-2) + 1 = -15 - 6 + 1 = -20. This matches the right-hand side of the equation, so it checks out! You should also verify the solution with the other two equations. This step helps catch any potential errors and confirms that you've correctly solved the system.

Conclusion: Mastering Systems of Equations

Solving systems of equations may seem complex at first, but by systematically applying methods like elimination, you can conquer these problems. Remember to always double-check your work to ensure accuracy. With practice, you'll become more comfortable with these techniques. Keep in mind that different systems may require slightly different strategies, but the fundamental principles remain the same. Continue practicing, and you'll become more confident in your abilities.

In this article, we've walked through the process of solving a 3x3 system of equations using the elimination method. By eliminating variables step by step, we were able to find the unique solution. Always remember to verify your results. So, go ahead and practice more problems, and you'll find that solving systems of equations becomes second nature!

For more in-depth practice and additional examples, consider exploring resources on Khan Academy https://www.khanacademy.org/.