Solving Systems Of Equations: A Step-by-Step Guide
Hey guys! Today, we're diving into the exciting world of solving systems of equations. Specifically, we'll tackle a system with three variables (x, y, and z). Don't worry, it might seem intimidating at first, but we'll break it down step by step so it's super easy to follow. We will convert the equations into simpler forms and solve it by using substitution or elimination methods. Ready to become equation-solving wizards? Let's jump in!
The Challenge: Our System of Equations
Before we get started, let's lay out the system of equations we're going to solve. This is our challenge for today:
- -3x + 12y + 6z = 78
- -3x + 3y - 4z = -4
- -4x - 2y + 3z = 17
Our goal is to find the values of x, y, and z that satisfy all three equations simultaneously. This solution will be represented as an ordered triple (x, y, z).
Why is this important? Solving systems of equations is a fundamental skill in mathematics and has applications in various fields like engineering, economics, and computer science. So, mastering this technique is definitely worth your time!
Step 1: Simplifying the First Equation
Okay, so looking at the equations, the first one, -3x + 12y + 6z = 78, seems like it could be simplified a bit. Notice that all the coefficients (-3, 12, and 6) are divisible by 3. So, let's divide the entire equation by -3. This keeps the equation balanced and makes the numbers smaller and more manageable.
Dividing by -3, we get:
x - 4y - 2z = -26
Let's call this simplified equation (1'). This is our new, improved first equation! Simplifying equations like this is a crucial strategy. It helps reduce complexity and makes the subsequent steps much smoother. Think of it as decluttering your workspace before starting a project – it makes everything easier to handle.
Why Simplify?
Simplifying equations isn't just about making the numbers smaller. It's about revealing the underlying structure of the equations. By dividing out the common factor, we've made the relationship between x, y, and z in the first equation clearer. This will be really helpful when we start combining equations to eliminate variables.
Also, smaller numbers mean less chance of making arithmetic errors. When you're dealing with multiple steps, even a small mistake can throw off your entire solution. So, simplifying is a smart move for accuracy as well.
Rewriting Simplified Equation
Now we can rewrite this equation to isolate x: x = 4y + 2z - 26. This form will be useful later when we use substitution to solve for the variables.
Step 2: Eliminating 'x' from Equations (2) and (3)
Alright, now that we've simplified the first equation, let's move on to the next crucial step: eliminating 'x' from equations (2) and (3). This is a classic technique in solving systems of equations, and it involves strategically combining equations to cancel out a variable.
Why eliminate a variable? Well, by eliminating 'x', we'll be left with two equations in just 'y' and 'z'. This makes the system much easier to solve because we'll have fewer unknowns in each equation. It's like simplifying a puzzle – by removing some pieces, the overall picture becomes clearer.
Eliminating 'x' from Equation (2)
First, let's work on eliminating 'x' from equation (2): -3x + 3y - 4z = -4. To do this, we'll use our modified equation (1'): x - 4y - 2z = -26. The key is to multiply equation (1') by -3 to match the coefficient of x in the original second equation.
Multiply equation (1') by -3:
-3(x - 4y - 2z) = -3(-26)
This gives us:
-3x + 12y + 6z = 78
Notice that the left side of this new equation is very similar to the left side of original equation 1.
Now, subtract the original equation (2) (-3x + 3y - 4z = -4) from this new equation:
(-3x + 12y + 6z) - (-3x + 3y - 4z) = 78 - (-4)
Simplifying this, we get:
9y + 10z = 82
Let's call this equation (4). We've successfully eliminated 'x' and now have an equation with just 'y' and 'z'!
Eliminating 'x' from Equation (3)
Now, let's eliminate 'x' from equation (3): -4x - 2y + 3z = 17. We'll use equation (1') again: x - 4y - 2z = -26. This time, we'll multiply equation (1') by -4 to match the coefficient of 'x' in equation (3).
Multiply equation (1') by -4:
-4(x - 4y - 2z) = -4(-26)
This gives us:
-4x + 16y + 8z = 104
Now, subtract equation (3) (-4x - 2y + 3z = 17) from this new equation:
(-4x + 16y + 8z) - (-4x - 2y + 3z) = 104 - 17
Simplifying, we get:
18y + 5z = 87
Let's call this equation (5). We've done it again! We've eliminated 'x' and now have another equation with just 'y' and 'z'.
Step 3: Solving for 'y' and 'z'
Great job, guys! We've made some serious progress. We now have two equations, (4) and (5), with only two unknowns, 'y' and 'z'. This is a much simpler system to solve.
Equation (4): 9y + 10z = 82
Equation (5): 18y + 5z = 87
We can use either substitution or elimination to solve for 'y' and 'z'. Let's use elimination again, as it seems like a good fit here.
Eliminating 'z'
To eliminate 'z', we need to make the coefficients of 'z' in equations (4) and (5) the same (but with opposite signs). The least common multiple of 10 and 5 is 10, so let's aim to make both coefficients 10 (or -10).
Multiply equation (5) by -2:
-2(18y + 5z) = -2(87)
This gives us:
-36y - 10z = -174
Now, add this modified equation (5) to equation (4):
(9y + 10z) + (-36y - 10z) = 82 + (-174)
Simplifying, we get:
-27y = -92
Now, divide both sides by -27 to solve for 'y':
y = -92/-27
y = 92/27
Awesome! We've found the value of 'y'!
Solving for 'z'
Now that we know 'y', we can plug it back into either equation (4) or (5) to solve for 'z'. Let's use equation (4): 9y + 10z = 82.
Substitute y = 92/27 into equation (4):
9 * (92/27) + 10z = 82
Simplify:
92/3 + 10z = 82
Subtract 92/3 from both sides:
10z = 82 - 92/3
10z = (246 - 92) / 3
10z = 154 / 3
Divide both sides by 10:
z = 154 / (3 * 10)
z = 77 / 15
Excellent! We've also found the value of 'z'!
Step 4: Solving for 'x'
We're almost there, guys! We've found 'y' and 'z', and now we just need to find 'x'. We can use any of our original equations (or the simplified version, equation (1')) to solve for 'x'. Let's use equation (1'): x - 4y - 2z = -26.
Substitute the values of 'y' and 'z' into equation (1'):
x - 4 * (92/27) - 2 * (77/15) = -26
Simplify:
x - 368/27 - 154/15 = -26
To get rid of the fractions, we need to find a common denominator for 27 and 15. The least common multiple of 27 and 15 is 135.
x - (368/27) * (5/5) - (154/15) * (9/9) = -26
x - 1840/135 - 1386/135 = -26
x - 3226/135 = -26
Add 3226/135 to both sides:
x = -26 + 3226/135
Find a common denominator:
x = (-26 * 135 + 3226) / 135
x = (-3510 + 3226) / 135
x = -284 / 135
There we have it! We've found the value of 'x'!
Step 5: The Solution
Woohoo! We did it! We've successfully solved the system of equations. Our solution is the ordered triple (x, y, z):
(-284/135, 92/27, 77/15)
Congratulations, guys! You've mastered a complex system of equations. Remember, the key is to break down the problem into smaller, manageable steps. Simplify when you can, eliminate variables strategically, and don't be afraid to tackle those fractions. With practice, you'll become a system-solving pro!
Tips for Success
Solving systems of equations can be challenging, but here are a few tips to help you succeed:
- Stay organized: Keep your work neat and clearly labeled. This will help you avoid mistakes and make it easier to track your progress.
- Double-check your work: Arithmetic errors are common, so take the time to check your calculations carefully.
- Practice makes perfect: The more you practice, the more comfortable you'll become with the techniques.
- Don't give up: Some systems of equations are tougher than others, but with persistence, you can solve them.
Conclusion
So, there you have it! We've walked through a detailed solution to a system of three equations with three variables. Remember, solving systems of equations is a valuable skill that can be applied in many different areas. Keep practicing, and you'll become a master at it! Keep up the amazing work, and I'll see you in the next math adventure!
