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

Hey guys! Today, we're diving into the world of solving systems of equations. This is a fundamental concept in algebra, and understanding it is super important. We'll break down how to find solutions, determine if there are infinitely many, and figure out if the equations are dependent. Ready to get started? Let's jump in!

Understanding Systems of Equations

Systems of equations are simply a set of two or more equations that we need to solve simultaneously. Each equation represents a relationship between variables, and the solution to the system is the set of values for those variables that satisfy all equations in the system. Think of it like finding the point (or points) where different lines or curves intersect on a graph. There are several methods we can use to solve these systems, and the best method often depends on the specific equations we are working with.

Let's talk about what it means to solve a system. When we solve a system of equations, we're looking for the values of the variables that make all the equations true at the same time. The solution is often represented as an ordered set of values, such as (x, y, z) if we have three variables. For instance, if a system has the solution (2, 3, 4), it means that plugging x = 2, y = 3, and z = 4 into every equation in the system will make those equations true. If a system has one solution, it's called a consistent system. If there is no solution, it's called an inconsistent system. In some cases, the equations in a system are dependent, which means one equation can be derived from the others. This often results in infinitely many solutions. Understanding these concepts is key to tackling problems involving systems of equations effectively. We are going to use a combination of substitution and elimination.

Before we proceed, let's clarify the various solution types we might encounter. First, we have a unique solution, where there's only one set of values for the variables that satisfy the equations. This is often the most straightforward case. Then there's the scenario where there are infinitely many solutions. This happens when the equations are dependent, meaning one or more equations are essentially duplicates or combinations of the others. Finally, there is no solution. This occurs when the equations are inconsistent, and there's no set of values for the variables that can make all equations true simultaneously. Recognizing these types of solutions will help us interpret our results correctly and understand the nature of the system we're working with. The goal is to find the values of x, y, and z that satisfy all three equations. Now, let's apply these concepts to solve the system of equations. Let's get to the steps!

Solving the Given System

Alright, let's get down to business and solve the system of equations you provided. We'll use a combination of elimination to systematically reduce the equations and find the values for x, y, and z. Keep your eyes peeled, as it might seem a bit challenging at first, but it's totally doable!

Here's our system:

6x - y + 3z = 10  (Equation 1)
x + 3y - z = 11  (Equation 2)
3x + 3y - 4z = 23 (Equation 3)

Step 1: Eliminate a Variable

Our goal is to reduce the system to simpler equations. Let's start by eliminating z from Equations 1 and 2. To do this, we'll multiply Equation 2 by 3 and then add it to Equation 1.

Multiply Equation 2 by 3:

3 * (x + 3y - z) = 3 * 11
3x + 9y - 3z = 33 (Equation 4)

Now, add Equation 4 to Equation 1:

(6x - y + 3z) + (3x + 9y - 3z) = 10 + 33
9x + 8y = 43 (Equation 5)

Step 2: Eliminate the Same Variable Again

Now, let's eliminate z again, this time using Equations 2 and 3. We'll multiply Equation 2 by -4 and add it to Equation 3.

Multiply Equation 2 by -4:

-4 * (x + 3y - z) = -4 * 11
-4x - 12y + 4z = -44 (Equation 6)

Add Equation 6 to Equation 3:

(3x + 3y - 4z) + (-4x - 12y + 4z) = 23 + (-44)
-x - 9y = -21 (Equation 7)

Step 3: Solve the Reduced System

Now we have two new equations (Equations 5 and 7) with just x and y. Let's solve them. First, multiply Equation 7 by 9:

-9x - 81y = -189 (Equation 8)

Now add Equation 5 and Equation 8:

(9x + 8y) + (-9x - 81y) = 43 + (-189)
-73y = -146
y = 2

Step 4: Back-Substitute

Substitute y = 2 into Equation 7 to solve for x:

-x - 9(2) = -21
-x - 18 = -21
-x = -3
x = 3

Then, substitute x = 3 and y = 2 into Equation 2 to solve for z:

3 + 3(2) - z = 11
3 + 6 - z = 11
9 - z = 11
-z = 2
z = -2

So, our solution is (x, y, z) = (3, 2, -2).

Solution and Conclusion

Alright! We've successfully solved the system of equations. We found a unique solution, which means the three equations intersect at a single point in 3D space. The solution is (x, y, z) = (3, 2, -2). This means if you substitute these values back into the original equations, they will all be true. Keep in mind that solving systems of equations is a fundamental skill in mathematics, with applications in a wide range of fields. Make sure you're comfortable with the steps we took: elimination, substitution, and back-substitution. With practice, you'll become a pro at tackling these problems. Always double-check your work, and never be afraid to ask for help if you need it. Keep practicing and you'll master it in no time! Congrats on making it through this problem, and keep up the fantastic work!