Solving Linear Equations: A Step-by-Step Guide

Hey guys! Let's dive into the world of solving linear equations, specifically the ones that look like this: 4x - 3y = -13 and 2x + 5y = 1. Sounds a bit intimidating, right? Don't worry, we'll break it down into easy-to-understand steps. This is a fundamental concept in algebra, and once you get the hang of it, you'll be able to solve a whole bunch of problems. We are going to find the solution to the system of equations. The solution to a system of equations is the set of values for the variables (in this case, x and y) that satisfy all equations in the system. The best part? There are multiple ways to tackle these problems, and we are going to explore a couple of methods.

The Substitution Method: Your Equation-Solving Superhero

Alright, first up, we have the substitution method. Think of it as your equation-solving superhero! The main idea here is to isolate one variable in one of the equations and then substitute that expression into the other equation. This transforms the problem into a single-variable equation, which is way easier to solve. Let's start with our equations again:

• Equation 1: 4x - 3y = -13
• Equation 2: 2x + 5y = 1

Step 1: Isolate a variable. Looking at the equations, it might be easiest to isolate x in Equation 2. Let's rearrange that one:

2x + 5y = 1 2x = 1 - 5y x = (1 - 5y) / 2

Step 2: Substitute. Now, we'll substitute this expression for x into Equation 1. Wherever we see x, we'll replace it with (1 - 5y) / 2:

4*((1 - 5y) / 2) - 3y = -13

Step 3: Solve for y. Time to simplify and solve for y. This is where your algebra skills come in handy:

2 * (1 - 5y) - 3y = -13 2 - 10y - 3y = -13 -13y = -15 y = 15 / 13

Step 4: Solve for x. Now that we know the value of y, we can plug it back into our expression for x that we found in Step 1:

x = (1 - 5 * (15 / 13)) / 2 x = (1 - 75 / 13) / 2 x = (-62 / 13) / 2 x = -31 / 13

So, using the substitution method, we've found that x = -31/13 and y = 15/13. This is the solution to our system of equations! These are the values that, when plugged into the original equations, will make both equations true. Pretty cool, huh? The substitution method is a solid approach for solving systems of linear equations. It's especially useful when one of the equations is already solved for a variable or can be easily rearranged. Remember to stay organized, double-check your work, and always substitute back into the original equations to verify your answer!

The Elimination Method: Wiping Out Variables

Next up, we have the elimination method, sometimes also called the addition method. The key here is to manipulate the equations so that when you add them together, one of the variables cancels out. It's like a magical disappearing act for variables! Let's get back to our original equations:

• Equation 1: 4x - 3y = -13
• Equation 2: 2x + 5y = 1

Step 1: Make coefficients match. We need to make the coefficients of either x or y match (but with opposite signs) so that they cancel out when we add the equations. Let's aim to eliminate x. We can multiply Equation 2 by -2:

-2 * (2x + 5y) = -2 * (1) -4x - 10y = -2

Step 2: Add the equations. Now, add the modified Equation 2 to Equation 1:

4x - 3y = -13 (-4x - 10y = -2)

Adding these gives us:

-13y = -15

Step 3: Solve for y. We can now easily solve for y:

y = 15 / 13

Step 4: Solve for x. Substitute the value of y back into either of the original equations. Let's use Equation 2:

2x + 5 * (15 / 13) = 1 2x + 75 / 13 = 1 2x = -62 / 13 x = -31 / 13

Voila! We got the same solution as before: x = -31/13 and y = 15/13. The elimination method is a super-efficient way to solve systems of equations, especially when the coefficients are already set up nicely. The beauty of this method lies in its ability to quickly eliminate a variable, leading directly to the solution. It's a fundamental technique for anyone learning algebra, and with practice, you'll become a pro at finding the right multipliers to eliminate the variables you want to remove. It is a slightly different approach but the same result.

Choosing the Right Method

So, which method is best? It really depends on the equations you're working with. Here's a quick guide:

• Substitution Method: Great when one of the equations is already solved for a variable or is easy to solve for a variable.
• Elimination Method: Best when the coefficients of one variable are easy to make match (or have opposite signs). This can often be the quicker method if the equations are set up in a way that allows for easy elimination.

In our example, both methods worked equally well. However, in other problems, one method might be clearly superior. It's good to be familiar with both methods so you can choose the one that's most efficient for the problem at hand.

Real-World Applications

Why should you care about solving systems of linear equations? Well, they pop up in all sorts of real-world situations! Here are a few examples:

• Economics: Determining the equilibrium point in supply and demand models.
• Finance: Calculating investment returns or managing budgets.
• Science: Solving for unknown variables in experimental data.
• Engineering: Designing structures or analyzing circuits.

Solving linear equations is a fundamental skill that opens the door to understanding and solving complex problems in various fields. It’s a core concept, and you can see it applied everywhere. From balancing your checkbook to calculating the trajectory of a rocket, understanding and being able to solve equations is a super useful skill. It's definitely worth the effort to master these methods!

Tips for Success

Here are some extra tips to help you succeed:

• Practice, practice, practice: The more you practice, the better you'll get. Work through various examples to build your confidence and understanding.
• Show your work: Always write out each step clearly. This helps you avoid mistakes and makes it easier to find and correct any errors.
• Check your answers: Always substitute your solutions back into the original equations to make sure they are correct.
• Stay organized: Keep your work neat and well-organized to avoid confusion.
• Don't be afraid to ask for help: If you get stuck, ask your teacher, a classmate, or search online for assistance. There are tons of resources available!

Conclusion

So, there you have it, guys! We've covered the basics of solving systems of linear equations using both the substitution and elimination methods. Remember, practice is key, and with time, you'll become a pro at solving these problems. Keep at it, and you'll be amazed at how quickly you improve. These methods are not just about finding answers; they are about developing a problem-solving mindset. You're building a foundation that will serve you well in all your future mathematical endeavors. Keep up the awesome work!