Solving Linear Equations: A Step-by-Step Guide

Hey guys! Ready to dive into some math? Today, we're going to break down how to solve a system of linear equations. Specifically, we'll tackle these two equations: 4x - 12y = 8 and -3x + 2y = 1. Don't worry if it sounds intimidating; we'll walk through it step-by-step to make it super clear. This is a fundamental concept in algebra, and understanding it will give you a solid foundation for more complex mathematical problems. We'll use a method called substitution to find the values of x and y that satisfy both equations simultaneously. So, grab your notebooks, and let's get started. By the end of this, you'll be solving these types of problems like a pro, and trust me, it's not as hard as it looks! This is a skill that comes in handy in all sorts of fields, from science and engineering to economics and even computer graphics. Are you ready to level up your math game? Let's do it!

Step 1: Isolate a Variable in One of the Equations

The first thing we need to do is pick one of the equations and solve for one of the variables, either x or y. Let's go with the second equation, -3x + 2y = 1, because it looks a bit simpler to manipulate. We'll solve for y in this case. To do this, we'll isolate y on one side of the equation. First, add 3x to both sides of the equation:

-3x + 2y + 3x = 1 + 3x

This simplifies to:

2y = 1 + 3x

Next, divide both sides by 2 to solve for y:

y = (1 + 3x) / 2

We've now expressed y in terms of x. This is a crucial step because it allows us to substitute this expression into the other equation. Always remember that the goal here is to get one variable by itself, which will help us solve for the other variable. Keep in mind, you can choose either equation to start with, and you can solve for either x or y – the key is consistency and accuracy in your algebraic manipulations. Make sure you double-check your work at each step to avoid any errors, and don't be afraid to take your time. This process is like solving a puzzle; each step brings you closer to the solution. Understanding this first step sets the stage for the rest of the problem, so make sure you've got it down before moving on. The ability to isolate a variable is a fundamental skill in algebra.

Step 2: Substitute the Expression into the Other Equation

Now that we have an expression for y ( y = (1 + 3x) / 2 ), we're going to substitute this into the other equation, which is 4x - 12y = 8. Wherever we see y in the equation, we'll replace it with (1 + 3x) / 2. This gives us:

4x - 12 * ((1 + 3x) / 2) = 8

Notice how we've eliminated y from the equation, leaving us with an equation that only contains x. The goal now is to solve for x. Simplify the equation. First, we can simplify -12 * ((1 + 3x) / 2) to -6 * (1 + 3x):

4x - 6(1 + 3x) = 8

Distribute the -6:

4x - 6 - 18x = 8

Combine like terms:

-14x - 6 = 8

Adding 6 to both sides:

-14x = 14

Finally, divide by -14 to solve for x:

x = -1

We've successfully solved for x! This substitution method is a powerful tool because it reduces the problem from two variables to one, making it solvable. Always double-check your substitution and simplification steps to catch any potential errors. Keep in mind that different problems might require you to substitute in x instead of y, or start with a different equation. The process is the same: find an expression for one variable and substitute it into the other equation. Remember, practice makes perfect! The more you do these types of problems, the more comfortable and efficient you will become.

Step 3: Solve for the Remaining Variable

We've found the value of x, which is -1. Now, we need to find the value of y. We can do this by substituting the value of x back into either of the original equations or, more conveniently, into the expression we found in Step 1: y = (1 + 3x) / 2. Substituting x = -1 into this equation, we get:

y = (1 + 3(-1)) / 2*

y = (1 - 3) / 2

y = -2 / 2

y = -1

So, we've found that y = -1. Now we have both values of x and y, and we are almost done! The substitution method works by systematically eliminating one variable, allowing you to solve for the other. This process is very important! When you're solving for y, make sure you're plugging the value of x into an equation that isolates y. It's also a good idea to substitute the values of x and y back into both of the original equations to check your work. If both equations are true with these values, then you know you've found the correct solution. Always double-check your calculations, especially the arithmetic, to minimize errors. By practicing this method, you will improve your problem-solving skills.

Step 4: Check Your Solution

It's always a good idea to verify your solution by plugging the values of x and y back into the original equations. This will help ensure that our solution is correct. Let's start with the first equation: 4x - 12y = 8. Substituting x = -1 and y = -1, we get:

4(-1) - 12(-1) = 8**

-4 + 12 = 8

8 = 8

This is true! Now, let's check the second equation: -3x + 2y = 1. Substituting x = -1 and y = -1, we get:

-3(-1) + 2(-1) = 1**

3 - 2 = 1

1 = 1

This is also true! Since both equations are true when we plug in x = -1 and y = -1, we can confidently say that this is the correct solution to the system of equations. Checking your solution is a critical step in problem-solving. It's like a quality control check; it helps you catch any mistakes you might have made during the process. This practice will not only help you get the right answers but also improve your overall understanding of the concepts. This step is about building confidence in your problem-solving abilities. Always double-check your work to make sure you're accurate.

Step 5: The Final Answer

Therefore, the solution to the system of equations 4x - 12y = 8 and -3x + 2y = 1 is x = -1 and y = -1. We found the values of x and y that satisfy both equations, meaning they are the point of intersection of the two lines represented by these equations. This entire process, from isolating variables to checking our solution, is a fundamental skill in algebra and is used extensively in many different fields. Congratulations, guys, you have successfully solved a system of linear equations! Now you can apply this knowledge to other systems of equations. Remember, practice is key. The more problems you solve, the more comfortable you'll become with the process. Keep exploring, keep learning, and don't be afraid to tackle challenging problems. Mathematics is a journey, and every step you take brings you closer to mastery. Good luck!