Solving Systems Of Equations: A Step-by-Step Guide
Have you ever stumbled upon a system of equations and felt completely lost? Don't worry, you're not alone! Systems of equations can seem intimidating at first, but with the right approach, they can be solved quite easily. In this comprehensive guide, we'll break down the process of solving a system of equations, step by step. We'll use the following system as our example:
4x + 3y - z = -6
6x - y + 3z = 12
8x + 2y + 4z = 6
So, let's dive in and learn how to conquer these mathematical puzzles!
Understanding Systems of Equations
Before we jump into solving, it's important to grasp the basic concept of a system of equations. A system of equations is simply a set of two or more equations containing the same variables. The goal is to find values for these variables that satisfy all equations simultaneously. Think of it like finding the common ground where all the equations agree. In our example, we have three equations with three unknowns (x, y, and z). The solution will be a set of values for x, y, and z that make all three equations true.
Systems of equations pop up in various real-world scenarios, from calculating mixtures in chemistry to modeling traffic flow. They're a fundamental tool in mathematics, science, and engineering. Mastering the art of solving them is a valuable skill. Different methods exist for tackling these systems, each with its own strengths and weaknesses. We'll focus on one of the most popular and versatile methods: elimination.
Why Elimination?
Elimination is a powerful technique that involves strategically manipulating the equations to eliminate one variable at a time. This simplifies the system, making it easier to solve. The beauty of elimination lies in its systematic approach, which can be applied to systems of any size. It's like a detective method, where we carefully gather clues and eliminate possibilities until we pinpoint the solution. The core idea is to add or subtract multiples of equations to cancel out variables. This might sound tricky, but it becomes clear with practice.
Before we start, let's rewrite our system of equations for clarity:
Equation 1: 4x + 3y - z = -6
Equation 2: 6x - y + 3z = 12
Equation 3: 8x + 2y + 4z = 6
Now, we're ready to roll up our sleeves and get to work!
Step 1: Choose a Variable to Eliminate
The first step in the elimination method is to choose a variable to eliminate. Looking at our system, we can see that the 'y' variable has coefficients 3, -1, and 2. It might be a good candidate for elimination because it's relatively easy to find multiples that will cancel out. However, the choice is yours! You could also choose to eliminate 'x' or 'z'. The process will work regardless, although some choices might lead to simpler calculations. To make our lives a bit easier, let's target the 'y' variable.
When deciding which variable to eliminate, consider the coefficients – the numbers multiplying the variables. Look for coefficients that are multiples of each other or have opposite signs. This will make the elimination process smoother. Remember, the goal is to create opposite coefficients for the chosen variable in two different equations. This way, when we add the equations, the variable will disappear. In our case, we can easily make the 'y' coefficients in Equation 2 and Equation 3 opposites of the 'y' coefficients in Equation 1.
It's also worth noting that sometimes, a little bit of foresight can save you time and effort. If you see a variable that has a coefficient of 1 or -1, it's often a good idea to target that variable for elimination. This is because it's easier to multiply an equation by a constant to match the coefficients. In our system, Equation 2 has a '-y' term, which means the coefficient is -1. This is another reason why eliminating 'y' is a sensible choice.
Preparing for Elimination
Before we can eliminate 'y', we need to manipulate our equations. We'll focus on Equations 1 and 2 first. Our aim is to make the 'y' coefficients opposites. Currently, they are 3 and -1. To make them opposites, we can multiply Equation 2 by 3. This will give us a '-3y' term, which will nicely cancel out the '3y' term in Equation 1. Remember, when we multiply an equation by a constant, we must multiply every term in the equation to maintain the equality.
So, let's multiply Equation 2 by 3:
3 * (6x - y + 3z) = 3 * 12
18x - 3y + 9z = 36
We'll call this new equation Equation 4. Now we have:
Equation 1: 4x + 3y - z = -6
Equation 4: 18x - 3y + 9z = 36
Notice that the 'y' coefficients are now 3 and -3. We're one step closer to eliminating 'y'!
Step 2: Eliminate the Chosen Variable
Now comes the fun part – eliminating 'y'! We have Equation 1 and Equation 4, where the 'y' coefficients are opposites. To eliminate 'y', we simply add these two equations together. When we add equations, we add the corresponding terms. The 'x' terms are added together, the 'y' terms are added together, the 'z' terms are added together, and the constants are added together.
Let's add Equation 1 and Equation 4:
(4x + 3y - z) + (18x - 3y + 9z) = -6 + 36
Combining like terms, we get:
22x + 0y + 8z = 30
22x + 8z = 30
Notice that the 'y' term has vanished! We've successfully eliminated 'y' and obtained a new equation with only two variables, 'x' and 'z'. Let's call this Equation 5:
Equation 5: 22x + 8z = 30
Eliminating 'y' Again
To solve for 'x' and 'z', we need another equation that involves only these two variables. We can achieve this by eliminating 'y' again, but this time using a different pair of equations. Let's use Equation 1 and Equation 3. The 'y' coefficients in these equations are 3 and 2. To eliminate 'y', we need to make these coefficients opposites. We can do this by multiplying Equation 1 by -2 and Equation 3 by 3.
First, let's multiply Equation 1 by -2:
-2 * (4x + 3y - z) = -2 * -6
-8x - 6y + 2z = 12
We'll call this Equation 6.
Next, let's multiply Equation 3 by 3:
3 * (8x + 2y + 4z) = 3 * 6
24x + 6y + 12z = 18
We'll call this Equation 7. Now we have:
Equation 6: -8x - 6y + 2z = 12
Equation 7: 24x + 6y + 12z = 18
The 'y' coefficients are now -6 and 6. Let's add these equations to eliminate 'y':
(-8x - 6y + 2z) + (24x + 6y + 12z) = 12 + 18
Combining like terms, we get:
16x + 14z = 30
This is our second equation with only 'x' and 'z'. Let's call it Equation 8:
Equation 8: 16x + 14z = 30
Now we have two equations (Equation 5 and Equation 8) with two unknowns ('x' and 'z'). We're making great progress!
Step 3: Solve the Reduced System
We've successfully reduced our original system of three equations with three unknowns to a system of two equations with two unknowns. This is a big step forward! Our reduced system is:
Equation 5: 22x + 8z = 30
Equation 8: 16x + 14z = 30
We can use the elimination method again to solve this system. Let's choose to eliminate 'x'. To do this, we need to make the 'x' coefficients opposites. We can multiply Equation 5 by -16 and Equation 8 by 22.
First, let's multiply Equation 5 by -16:
-16 * (22x + 8z) = -16 * 30
-352x - 128z = -480
We'll call this Equation 9.
Next, let's multiply Equation 8 by 22:
22 * (16x + 14z) = 22 * 30
352x + 308z = 660
We'll call this Equation 10. Now we have:
Equation 9: -352x - 128z = -480
Equation 10: 352x + 308z = 660
The 'x' coefficients are now opposites. Let's add these equations to eliminate 'x':
(-352x - 128z) + (352x + 308z) = -480 + 660
Combining like terms, we get:
180z = 180
Dividing both sides by 180, we find:
z = 1
We've found the value of 'z'! Now we can substitute this value back into either Equation 5 or Equation 8 to solve for 'x'. Let's use Equation 5:
22x + 8z = 30
22x + 8 * 1 = 30
22x + 8 = 30
22x = 22
x = 1
We've also found the value of 'x'! Now we have x = 1 and z = 1.
Step 4: Back-Substitute to Find the Remaining Variable
We've found the values of 'x' and 'z'. Now we need to find the value of 'y'. To do this, we can substitute the values of 'x' and 'z' back into any of our original equations (Equation 1, Equation 2, or Equation 3). Let's use Equation 1:
4x + 3y - z = -6
4 * 1 + 3y - 1 = -6
4 + 3y - 1 = -6
3y + 3 = -6
3y = -9
y = -3
We've found the value of 'y'! So, the solution to our system of equations is x = 1, y = -3, and z = 1.
Step 5: Check Your Solution
It's always a good idea to check your solution to make sure it's correct. To do this, we substitute the values of x, y, and z back into all three of our original equations. If the equations hold true, then our solution is correct.
Let's check Equation 1:
4x + 3y - z = -6
4 * 1 + 3 * -3 - 1 = -6
4 - 9 - 1 = -6
-6 = -6
Equation 1 holds true.
Let's check Equation 2:
6x - y + 3z = 12
6 * 1 - (-3) + 3 * 1 = 12
6 + 3 + 3 = 12
12 = 12
Equation 2 holds true.
Let's check Equation 3:
8x + 2y + 4z = 6
8 * 1 + 2 * -3 + 4 * 1 = 6
8 - 6 + 4 = 6
6 = 6
Equation 3 holds true. Since our solution satisfies all three equations, we can confidently say that it is correct!
Conclusion
Solving systems of equations might seem daunting at first, but by following a systematic approach like elimination, you can break down the problem into manageable steps. Remember to choose a variable to eliminate, manipulate the equations to create opposite coefficients, add the equations to eliminate the variable, and repeat the process until you have a single equation with one unknown. Then, back-substitute to find the values of the other variables. And most importantly, always check your solution!
With practice, you'll become a pro at solving systems of equations. These skills are invaluable in various fields, so keep honing your mathematical prowess! If you're looking for more resources on systems of equations, check out this helpful guide on Khan Academy. Happy solving!
