Solving Equations: A Step-by-Step Guide
Hey everyone! Today, we're diving into the world of solving systems of linear equations. This might sound a bit intimidating at first, but trust me, it's like a puzzle, and we'll break it down step by step to make it super easy. We're going to tackle a specific set of equations: x + 2y + 3z = 32, 2x - y + 4z = –7, and -3x + y - z = 4. Our goal is to find the values of x, y, and z that satisfy all three equations simultaneously. Think of it like finding a secret code that unlocks the solution! We'll use a method called elimination, which is like a mathematical magic trick that simplifies the equations until we can easily find the values. This method involves strategically adding or subtracting the equations to eliminate one variable at a time. By the end, you'll be able to solve similar problems with confidence. It's all about logical thinking and following a clear set of steps, so let's get started. Get ready to flex those math muscles and discover how to solve these equations like a pro! This process is not just about finding answers; it's about developing critical thinking and problem-solving skills that are valuable in all aspects of life. So, buckle up, and let's unravel the mystery of these equations together. The key to mastering this is practice, practice, and more practice. The more you work through different examples, the more comfortable and proficient you will become. Remember, everyone learns at their own pace, so don't get discouraged if it takes a little time to grasp everything. The important thing is to keep trying and to celebrate each small victory along the way. We'll be using this elimination method in order to get our values, let's explore this method in detail.
Understanding the Elimination Method
Alright, let's talk about the elimination method, the star of our show. The basic idea is to manipulate the equations in a way that allows us to eliminate one variable at a time. This is done by adding or subtracting the equations. We want to get rid of a variable so we can solve for the others. Think of it like this: if you have two equations, you can add or subtract them to create a new equation that has fewer variables. The goal is to create a situation where, when you add or subtract the equations, one of the variables cancels out. This can happen if the coefficients of a variable are the same but with opposite signs (e.g., +y and -y) or if the coefficients are the same. If the coefficients aren't already set up for easy elimination, we can multiply one or both equations by a constant to make the coefficients match. Then, we can add or subtract the equations to eliminate that variable. Once we've eliminated one variable, we are left with an equation with just two variables, which we can then solve. We will repeat this process until we have an equation with only one variable, which is easily solved. This will give us the value of one variable. Now, this one value can be plugged into any of the previous equations to solve for other variables. Let's get our hands dirty by applying the method to our equations!
Step-by-Step Solution
Let's get down to the nitty-gritty and solve our equations: x + 2y + 3z = 32, 2x - y + 4z = –7, and -3x + y - z = 4. We'll methodically work through each step to find the values of x, y, and z. Get ready to follow along as we apply the elimination method to solve the mystery of these equations, okay? Here we go! First, let's label our equations for easy reference:
- Equation 1: x + 2y + 3z = 32
- Equation 2: 2x - y + 4z = –7
- Equation 3: -3x + y - z = 4
Our first step is to eliminate one variable. Let's eliminate y. Notice that in Equation 2, the coefficient of y is -1, and in Equation 3, it's +1. This makes our job easier! We can add Equation 2 and Equation 3 directly to eliminate y. Doing so, we get:
(2x - y + 4z) + (-3x + y - z) = -7 + 4
This simplifies to:
-x + 3z = -3 (Let's call this Equation 4)
Now we have a new equation with only x and z. Next, we'll work with Equations 1 and 3 to eliminate y again. To do this, we'll multiply Equation 3 by -2 so that the y terms cancel out when we add the equations. So, multiplying Equation 3 by -2 gives us:
6x - 2y + 2z = -8
Now, add this new equation to Equation 1:
(x + 2y + 3z) + (6x - 2y + 2z) = 32 + (-8)
This simplifies to:
7x + 5z = 24 (Let's call this Equation 5)
Now we have two equations with two variables: Equation 4 (-x + 3z = -3) and Equation 5 (7x + 5z = 24). Let's eliminate x from these two equations. Multiply Equation 4 by 7:
-7x + 21z = -21
Now, add this to Equation 5:
(7x + 5z) + (-7x + 21z) = 24 + (-21)
This simplifies to:
26z = 3
Divide both sides by 26:
z = 3/26
We found z! Now that we have the value of z, we can substitute it into one of our equations to find x. Let's use Equation 4:-x + 3z = -3
Substitute z = 3/26: -x + 3(3/26) = -3*
-x + 9/26 = -3
-x = -3 - 9/26 = -87/26
x = 87/26
And finally, we have x! Now that we have x and z, we can substitute these values into any of our original equations to find y. Let's use Equation 3: -3x + y - z = 4
Substitute x = 87/26 and z = 3/26:
-3(87/26) + y - (3/26) = 4*
-261/26 + y - 3/26 = 4
y = 4 + 264/26 = 104/26 + 264/26 = 368/26 = 184/13
And we have y! So, the solution to our system of equations is: x = 87/26, y = 184/13, and z = 3/26. Pretty cool, right? We solved the puzzle!
Checking Your Answers and Further Practice
Great job, everyone! You've successfully navigated the elimination method to solve a system of linear equations. Now, the next crucial step is to check your answers. Always, always, always make sure your solution is correct. Checking your answers is like double-checking your work on a test – it helps you catch any mistakes and ensures you have the correct solution. To check your answers, substitute the values of x, y, and z back into the original equations. If the equations hold true, then your solution is correct. This is how it's done: take the calculated values for x, y, and z and put them in the initial equations, let's see if our solutions work for the first equation x + 2y + 3z = 32. Replacing x=87/26, y=184/13 and z=3/26 we have:
87/26 + 2(184/13) + 3(3/26) = 32
87/26 + 368/13 + 9/26 = 32
(87+736+9)/26 = 32
832/26=32
32=32
Correct! Let's check with the second equation: 2x - y + 4z = –7.
2(87/26) - 184/13 + 4(3/26) = –7
174/26 - 184/13 + 12/26 = –7
(174-368+12)/26 = –7
-182/26 = -7
-7=-7
Also correct! Finally, with our third equation: -3x + y - z = 4
-3(87/26) + 184/13 - 3/26 = 4
-261/26 + 184/13 - 3/26 = 4
(-261+368-3)/26 = 4
104/26=4
4=4
Awesome, our solution is correct! To solidify your understanding, the best thing to do is practice. Try solving different systems of equations on your own. You can find plenty of practice problems online or in your textbook. Start with easier problems and gradually move to more complex ones. Practice helps build confidence and proficiency. Don’t hesitate to revisit the steps we covered, and don’t be afraid to ask for help if you get stuck. Keep practicing, and you will become a pro in no time! Remember, practice makes perfect!