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

Hey there, math enthusiasts! Today, we're diving into the world of linear equations and learning how to solve systems of these equations. Linear equations are the building blocks of many mathematical models and understanding how to solve them is a fundamental skill. We'll walk through a specific example, breaking down each step to make the process super clear. So, grab your pencils and let's get started!

We'll be tackling the following system of linear equations:

$\begin{array}{l} -3 x-2 y-2 z=-15 \\ -3 x-y-2 z=-13 \\ -x+y-z=-3 \end{array}$

This system involves three equations with three unknowns (x, y, and z). Our goal is to find the values of x, y, and z that satisfy all three equations simultaneously. There are several methods to solve these systems, and we'll employ a combination of elimination and substitution to find the solution. Let's start with the first two equations to eliminate 'x'. This is a common and effective strategy.

Step 1: Eliminate 'x' from the First Two Equations

Our initial step is to eliminate 'x' from the first two equations. The elimination method involves manipulating the equations to cancel out one of the variables. In this case, we can subtract the second equation from the first equation to eliminate 'x'.

Here's how it looks:

• Equation 1: -3x - 2y - 2z = -15
• Equation 2: -3x - y - 2z = -13

Subtract Equation 2 from Equation 1:

(-3x - 2y - 2z) - (-3x - y - 2z) = -15 - (-13)

Simplifying this, we get:

-y = -2

Therefore, y = 2.

Awesome, we've found the value of y! Now, let's substitute this value into one of the original equations to simplify things further. Substituting this value helps reduce the number of variables in the equation, making it easier to solve. You'll see how this makes the next steps much more manageable. This is a classic example of how solving linear equations is really just a puzzle, and each step you take brings you closer to the solution. The more you practice, the faster and more intuitive this process becomes. We'll proceed with substituting the value of 'y' into the remaining equations to solve for other variables. This systematic approach is key to successfully navigating through complex problems. Keep going, you're doing great!

Step 2: Substitute y = 2 into Equations

Now that we know y = 2, we can substitute this value into any of the original equations. Let's substitute y = 2 into the second and third equations, as these look a little simpler to work with. Remember, the choice is yours, and you should pick whatever looks easiest to you.

• Equation 2: -3x - y - 2z = -13 becomes -3x - 2 - 2z = -13
• Equation 3: -x + y - z = -3 becomes -x + 2 - z = -3

Simplifying these, we get:

• -3x - 2z = -11 (Equation 4)
• -x - z = -5 (Equation 5)

Great, we now have two new equations with only two variables, 'x' and 'z'. This is a much easier problem to tackle. The goal here is to reduce the number of variables one by one until you can solve them independently. This stepwise reduction is a core strategy in algebra. Let's move on to the next step, where we will solve for 'x' and 'z'. Don't worry, we are almost there! Keep up the good work.

Step 3: Solve for x and z

Now we're going to solve the new system of equations (Equations 4 and 5) to find the values of 'x' and 'z'.

• Equation 4: -3x - 2z = -11
• Equation 5: -x - z = -5

We can use the elimination method again. Let's multiply Equation 5 by -2, and then subtract the result from Equation 4. This will help us eliminate 'z'. This type of manipulation is common in algebra and essential for problem-solving.

Multiply Equation 5 by -2:

2x + 2z = 10

Now, add this modified equation to Equation 4:

(-3x - 2z) + (2x + 2z) = -11 + 10

Simplifying, we get:

-x = -1

Therefore, x = 1.

Now that we know x = 1, we can substitute this value into Equation 5 to find 'z'.

-1 - z = -5

Solving for z, we get:

z = 4

We've found the values for x and z. We are almost at the end! Remember the steps: eliminate, substitute, and solve! That is it!

Step 4: Final Solution

We have now determined the values for x, y, and z:

• x = 1
• y = 2
• z = 4

Therefore, the solution to the system of linear equations is (1, 2, 4). This means that when we substitute these values into the original equations, they will all be true.

This is the final answer! Great job!

Conclusion: Mastering Linear Equations

Congrats, you made it through! Solving systems of linear equations might seem daunting at first, but with a systematic approach and practice, it becomes much easier. We've used elimination and substitution to find the solution. Remember, the key is to eliminate one variable at a time until you can solve for each variable individually. Keep practicing, and you'll become a pro at solving these types of problems. Remember, the skills you learn here are fundamental to various fields. From engineering to economics, understanding and solving these kinds of problems is useful. So keep up the great work and happy solving! Do not hesitate to repeat the process.

If you liked this article, please share it with your friends! Do not forget to practice, practice, practice! See you in the next one!