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

Hey guys! Ever stumbled upon a system of equations and felt like you're staring at a mathematical monster? Don't worry, we've all been there! Solving systems of equations might seem daunting at first, but with the right approach, it can become a piece of cake. In this article, we'll break down a specific system of equations step-by-step, making it super easy to understand. So, grab your pencils, and let's dive in!

Understanding the System of Equations

Before we jump into the solution, let's first understand what we're dealing with. A system of equations is a set of two or more equations containing the same variables. The goal is to find the values of these variables that satisfy all equations simultaneously. In simpler terms, we're looking for the numbers that, when plugged into the equations, make them all true. Think of it like a puzzle where each equation is a clue, and we need to find the combination that unlocks the solution.

The system we'll be tackling today is:

2x - 3y + 2z = -4
3y - 5z = -15
4z = 12

Notice that this system has three equations and three variables (x, y, and z). This type of system is often solvable, meaning we can find unique values for x, y, and z that work for all three equations. The structure of this particular system is quite helpful because the last equation only involves z, allowing us to solve for it directly. This is a classic example of a system that can be solved using a method called back-substitution, which we'll explore in detail.

Step 1: Solve for z

The cornerstone of solving this system lies in recognizing the simplicity of the third equation: 4z = 12. This equation only involves the variable z, making it straightforward to isolate and solve. To find the value of z, we simply need to divide both sides of the equation by 4. This isolates z on one side, revealing its numerical value.

Let's perform the calculation:

4z = 12
z = 12 / 4
z = 3

Therefore, the value of z is 3. This is our first breakthrough! With the value of z in hand, we can now move on to the next step, which involves using this newfound knowledge to solve for another variable. The beauty of this method, called back-substitution, is that it allows us to progressively unravel the system, one variable at a time. Now that we know z, we can substitute its value into the second equation to solve for y. This step-by-step approach is what makes solving systems of equations manageable and, dare I say, even enjoyable!

Step 2: Solve for y

Now that we've successfully determined the value of z to be 3, we can leverage this information to find the value of y. This is where the magic of back-substitution truly shines. We'll take the value of z and plug it into the second equation of our system:

3y - 5z = -15

By substituting z = 3, we transform this equation into one that only involves y, making it solvable. Let's perform the substitution:

3y - 5(3) = -15
3y - 15 = -15

Now, we have a simple equation in terms of y. To isolate y, we first add 15 to both sides of the equation:

3y - 15 + 15 = -15 + 15
3y = 0

Finally, we divide both sides by 3 to solve for y:

3y / 3 = 0 / 3
y = 0

So, we've found that y = 0. This is another significant step forward. We now know the values of both z and y. The final piece of the puzzle is to find the value of x. We'll use the same back-substitution technique, but this time, we'll plug the values of both y and z into the first equation.

Step 3: Solve for x

With the values of z and y now known (z = 3 and y = 0), we're in the home stretch! The final variable we need to solve for is x. To do this, we'll use the first equation in our system:

2x - 3y + 2z = -4

We'll substitute the values we found for y and z into this equation:

2x - 3(0) + 2(3) = -4

Simplifying the equation, we get:

2x + 6 = -4

To isolate x, we first subtract 6 from both sides:

2x + 6 - 6 = -4 - 6
2x = -10

Finally, we divide both sides by 2 to solve for x:

2x / 2 = -10 / 2
x = -5

Therefore, the value of x is -5. We've successfully found the values for all three variables! x = -5, y = 0, and z = 3. Now, the crucial step is to verify our solution to ensure it satisfies all three equations in the original system.

Step 4: Verify the Solution

We've arrived at a potential solution: x = -5, y = 0, and z = 3. But before we declare victory, it's crucial to verify that these values actually satisfy all three equations in our original system. This step ensures that we haven't made any errors along the way and that our solution is correct. Think of it as the final boss battle in our equation-solving quest!

Let's plug these values into each equation and see if they hold true:

Equation 1: 2x - 3y + 2z = -4

2(-5) - 3(0) + 2(3) = -4
-10 - 0 + 6 = -4
-4 = -4  (True!)

Equation 2: 3y - 5z = -15

3(0) - 5(3) = -15
0 - 15 = -15
-15 = -15  (True!)

Equation 3: 4z = 12

4(3) = 12
12 = 12  (True!)

Hooray! Our values for x, y, and z satisfy all three equations. This confirms that our solution is correct. We've successfully navigated the system of equations and emerged victorious!

Conclusion

So there you have it, guys! We've successfully solved the system of equations using the method of back-substitution. Remember, the key is to break down the problem into smaller, manageable steps. By first solving for z, then substituting that value to solve for y, and finally using both z and y to solve for x, we were able to unravel the mystery. And most importantly, we verified our solution to ensure accuracy. Solving systems of equations might seem intimidating at first, but with practice and a systematic approach, you'll become a pro in no time. Keep practicing, and don't be afraid to tackle those mathematical monsters head-on!