Solving Equations: A Step-by-Step Guide

Oct 25, 2025 by SLV Team 40 views

Hey everyone! Today, we're diving into the world of solving systems of linear equations. It might sound a bit intimidating at first, but trust me, with a little practice, you'll be cracking these problems like a pro. We'll break down the process step by step, using the example you provided:

$x + 2y + z = 4$ $2x + y - z = -1$ $x - y - z = -2$

So, grab your pens and paper, and let's get started. We are going to find out how to solve this system of linear equations, which is a set of equations where the highest power of the variables is 1. We will use a method called elimination to solve this. The goal is to eliminate variables one by one until we are left with a single variable that we can solve for. After that, we'll substitute the value we found back into the equations to find the values of the other variables. This is like a mathematical treasure hunt, and we're the explorers! Ready to uncover the secrets of these equations? Let's go!

Step 1: Choosing Equations and Variables

Okay, guys, the first thing we need to do is pick which equations we want to work with. There isn't a single 'right' way to do this; it's more about looking for the easiest path. Look for variables that have coefficients that are easy to manipulate – like a +1 or -1. In our system, the 'z' variable looks like a good place to start because we have both positive and negative 'z' terms. This will make the elimination process simpler. Let's take a closer look at our equations again:

$x + 2y + z = 4$
$2x + y - z = -1$
$x - y - z = -2$

We will start by eliminating 'z' using equations 1 and 2. Then, let's eliminate 'z' again but using equations 2 and 3. This approach will give us two new equations, each with only 'x' and 'y' variables. We can then solve this new system of two equations to find the values of 'x' and 'y'. This strategy, where you simplify the problem step-by-step, is commonly used in mathematics and makes the overall problem much more manageable.

Now, let's proceed with the first part of this elimination process. We will rewrite the equations we chose so that we can clearly see them. By adding equations 1 and 2, we can eliminate 'z' directly because the 'z' terms have opposite signs. We do not need to multiply any equations by a constant. This is the beauty of this system, it's ready to be used!

Adding equations 1 and 2: $(x + 2y + z) + (2x + y - z) = 4 + (-1)$

Simplifying this, we get: $3x + 3y = 3$

Now, we move on to the second part where we will eliminate 'z' using equations 2 and 3. The aim here is still the same: to create a new equation that only contains 'x' and 'y', making the solution of the system more manageable. Notice that the 'z' terms in equations 2 and 3 are both negative. To eliminate 'z', we need to change one of the 'z' coefficients to its opposite. We can do this by multiplying equation 3 by -1. This step is important because it sets the stage for the 'z' terms to cancel each other out when we add the equations. It's a clever trick, right?

Multiplying equation 3 by -1: $-1 * (x - y - z) = -1 * (-2)$

Which simplifies to: $-x + y + z = 2$

Now, adding the modified equation 3 to equation 2: $(2x + y - z) + (-x + y + z) = -1 + 2$

Simplifying the result: $x + 2y = 1$

By following these steps, we have successfully created two new equations where the variable 'z' has been eliminated, paving the way for us to solve the system further.

Why Eliminate Variables?

Eliminating variables is a strategic approach in solving systems of equations because it simplifies the problem. By systematically removing variables, we reduce the complexity of the system. For instance, when we eliminate 'z' from our three equations, we're left with equations that involve only 'x' and 'y'. This simplifies the solving process because we now have fewer unknowns to deal with at each step. This method makes the equations easier to manipulate and solve.

Elimination is particularly helpful when you have a system with several variables, such as three variables in our case. It breaks down a complex problem into smaller, more manageable parts. By eliminating variables one at a time, we gradually narrow down our unknowns until we can isolate and solve for each variable. This method is a cornerstone in algebra, making it a crucial tool for anyone dealing with mathematical models or real-world problem-solving.

Step 2: Solving the New System of Equations

Alright, folks, now that we've done the hard work of eliminating 'z', we've got a new system of two equations with two variables:

$3x + 3y = 3$
$x + 2y = 1$

Our mission now is to solve this system for 'x' and 'y'. Let's do this by eliminating one of the variables again. I see a chance to make this easy. We can isolate 'x' from equation 2 and substitute it into equation 1. Let's make 'x' the subject in equation 2.

Isolating 'x' from equation 2: $x = 1 - 2y$

Now, we will substitute this expression for 'x' into equation 1: $3 * (1 - 2y) + 3y = 3$

Then, simplifying the substitution: $3 - 6y + 3y = 3$

Combine the 'y' terms: $3 - 3y = 3$

Subtracting 3 from both sides: $-3y = 0$

Finally, dividing by -3: $y = 0$

Great! We have found the value of y. Now that we have the value of 'y', we can plug it into any of the equations with both 'x' and 'y' to find the value of 'x'. We are going to use the result we obtained while preparing the substitution earlier. Remember, $x = 1 - 2y$ . Let's plug the value of y into this equation:

$x = 1 - 2 * 0$ $x = 1$

So, we've found that $x = 1$ and $y = 0$ . We're making great progress, aren't we?

Why Substitution Works

Substitution is a powerful method because it allows us to reduce the number of variables in an equation. When we find an expression for one variable in terms of another, we can replace that variable in another equation with the expression. This turns an equation with two variables into one with just one variable, which we can solve directly. This is extremely helpful, especially in complex systems.

In our current example, after isolating 'x' from one equation, substituting it into the other equation reduced the problem to just 'y'. This simplification allows us to solve the problem step-by-step.

The cool thing about substitution is its flexibility. It doesn’t matter which equation you start with or which variable you choose to isolate; the process is the same. The key is to strategically choose the equation and variable that will make the substitution the simplest. This is another fundamental concept in algebra and is used extensively in fields like engineering and economics.

Step 3: Finding the Remaining Variable

We're in the final stretch, guys! We have $x = 1$ and $y = 0$ . The last step is to use these values to find 'z'. We can plug these values back into any of the original equations that include 'z'. I think we can use the first equation, it will make our calculations easy. Remember our first equation?

$x + 2y + z = 4$

Now, let's substitute the values of x and y: $1 + 2 * 0 + z = 4$

This simplifies to: $1 + z = 4$

Subtracting 1 from both sides: $z = 3$

And there we have it! We've found all the values!

$x = 1$
$y = 0$
$z = 3$

The Importance of Back-Substitution

Back-substitution is the final act of this process. It involves substituting the values we've found back into the equations to solve for the remaining variables. This stage is crucial because it connects all the steps of our solution, ensuring that the values we’ve calculated are consistent with the original system of equations. In essence, back-substitution is like a validation step. It allows us to confirm that the values of the variables simultaneously satisfy all the equations in the system.

The process of back-substitution not only helps in finding the final values but also helps in spotting any errors that may have occurred during the previous steps. By checking whether our solutions satisfy all equations, we can increase our confidence in the accuracy of our final result. This is a very valuable skill in problem-solving in general.

Step 4: Verification

Alright, let's make sure our solutions are correct by plugging our values back into the original equations. We'll use the original three equations to check:

$x + 2y + z = 4$
$2x + y - z = -1$
$x - y - z = -2$

Using our values $x = 1$ , $y = 0$ , and $z = 3$ , let's check each equation:

Equation 1: $1 + 2(0) + 3 = 4$ $1 + 0 + 3 = 4$ $4 = 4$ (Correct!)
Equation 2: $2(1) + 0 - 3 = -1$ $2 + 0 - 3 = -1$ $-1 = -1$ (Correct!)
Equation 3: $1 - 0 - 3 = -2$ $-2 = -2$ (Correct!)

All three equations check out! This means our solution is correct. We successfully solved the system of linear equations.

Verification: The Crucial Check

Verification is an incredibly important step in solving any mathematical problem. It involves confirming that the solutions we’ve found satisfy all conditions and constraints of the original problem. This step helps us catch any errors in our calculations and ensures we have the correct answer. The best part is that it is quite easy to do. You just need to substitute all your values into each equation and check if the result is correct.

In our case, by substituting the values of 'x', 'y', and 'z' back into the original equations, we verified that they hold true. This confirms that our solution satisfies all equations simultaneously. Not only does this validate our process, but it also provides us with the confidence to know we did it right. The act of verification is essential in many fields, from science to engineering. It assures the correctness of a solution and prevents further errors.

Conclusion

Great job, everyone! We've successfully solved a system of linear equations. Remember, the key is to take it step by step, choosing the right method (elimination or substitution), and always verifying your answer. Keep practicing, and you'll become a pro at this. Keep learning and pushing your limits. You got this!