Solving 2x+3y=3 And 3x-y=10 With Substitution Method

Hey guys! Have you ever stumbled upon a system of equations and felt like you were lost in a maze? Don't worry, you're not alone! Many people find solving simultaneous equations a bit tricky, but with the right method and a little practice, you'll be a pro in no time. In this article, we're going to dive deep into one of the most effective techniques: the substitution method. We'll break it down step-by-step, using the equation 2x + 3y = 3 and 3x - y = 10 as our example. So, grab your pencils, and let's get started!

Understanding Systems of Equations

Before we jump into the substitution method, let's make sure we're all on the same page about what a system of equations actually is. Simply put, a system of equations is a set of two or more equations that share the same variables. The goal is to find values for these variables that satisfy all equations simultaneously. Think of it like a puzzle where you need to find the pieces that fit perfectly into multiple spots at once.

These systems pop up everywhere, from simple algebra problems to complex real-world scenarios. Whether you're calculating the break-even point for your small business or modeling the trajectory of a rocket, understanding systems of equations is a crucial skill. So, mastering methods like substitution is not just about acing your math test; it's about equipping yourself with a powerful tool for problem-solving in various aspects of life.

Why is the substitution method so useful? Well, it's particularly handy when one of the equations can be easily rearranged to isolate a single variable. This makes it much easier to plug that expression into the other equation and solve for the remaining variable. It's like finding a shortcut through the maze, saving you time and effort. So, let's get into the nitty-gritty of how it works.

The Substitution Method: A Step-by-Step Approach

The substitution method is a neat way to solve systems of equations by, you guessed it, substituting one equation into another. Here’s how it works, step by step:

Step 1: Isolate a Variable in One Equation

The first step is to pick one of the equations and solve it for one of the variables. Look for the equation where it’s easiest to isolate a variable – usually, this means finding a variable with a coefficient of 1 or -1. This simplifies the process and reduces the chance of making mistakes with fractions. In our example, we have:

• 2x + 3y = 3
• 3x - y = 10

Looking at these equations, it seems easier to isolate y in the second equation (3x - y = 10). To do this, we can add y to both sides and subtract 10 from both sides. This gives us:

3x - 10 = y

Or, more commonly written:

y = 3x - 10

See how we managed to get y all by itself on one side? That's exactly what we want. We've now successfully isolated y in the second equation. This step is crucial because it sets the stage for the substitution part of the method. By isolating a variable, we've created an expression that we can plug into the other equation, effectively reducing the system to a single equation with one variable. This makes the problem much easier to solve. So, always take a moment to identify the easiest variable to isolate – it can save you a lot of headaches down the road.

Step 2: Substitute the Expression into the Other Equation

Now comes the fun part – the actual substitution! We take the expression we found in Step 1 and plug it into the other equation. Remember, we isolated y in the second equation (y = 3x - 10). So, we'll substitute this expression for y in the first equation (2x + 3y = 3).

This means we replace y in the first equation with (3x - 10). So, 2x + 3y = 3 becomes:

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

Notice how we've replaced y with the entire expression (3x - 10)? It's super important to put the expression in parentheses to make sure you distribute the multiplication correctly. This is a common spot where people make mistakes, so pay close attention to this detail. Now, we have an equation with only one variable, x. This is a huge step forward because we can now solve for x directly. The substitution has transformed our system of two equations into a single equation that's much easier to handle. It's like turning a complex puzzle into a simpler one. So, make sure you understand this substitution step thoroughly – it's the heart of the substitution method.

Step 3: Solve for the Remaining Variable

We've done the hard part – now it's just a matter of solving the equation we created in Step 2. Remember, after substituting, we got:

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

First, we need to distribute the 3 across the parentheses:

2x + 9x - 30 = 3

Next, combine like terms (the x terms):

11x - 30 = 3

Now, we want to isolate the x term. Add 30 to both sides of the equation:

11x = 33

Finally, divide both sides by 11 to solve for x:

x = 3

Yay! We've found the value of x! This is a significant milestone. We've successfully navigated the substitution and simplification steps to uncover one of the variables in our system. This value of x is a key piece of the puzzle. But remember, we're not done yet. We still need to find the value of y. The good news is that now that we know x, finding y is much easier. We'll use this value in the next step to complete our solution.

Step 4: Substitute the Value Back to Find the Other Variable

We're on the home stretch! We know that x = 3. Now, we need to find the value of y. To do this, we can substitute the value of x back into either of the original equations or the equation we got when we isolated y in Step 1 (y = 3x - 10). It's usually easiest to use the isolated equation, as it's already set up to solve for y.

So, let's plug x = 3 into y = 3x - 10:

y = 3(3) - 10

Now, just simplify:

y = 9 - 10

y = -1

And there we have it! We've found that y = -1. This is the final piece of the puzzle. We now know the values of both x and y that satisfy both equations in our system. This step is crucial because it completes the solution. Without finding both variables, we haven't truly solved the system. So, always remember to substitute the value you found back into one of the equations to find the remaining variable. It's the last step in the substitution method, and it brings us to the complete answer.

Step 5: Check Your Solution

Before we declare victory, it’s always a good idea to double-check our solution. This is a crucial step to ensure we haven't made any mistakes along the way. To check, we substitute the values we found for x and y into both original equations. If the values satisfy both equations, we know we've got the correct solution.

We found that x = 3 and y = -1. Let's plug these values into our original equations:

• Equation 1: 2x + 3y = 3 2(3) + 3(-1) = 3 6 - 3 = 3 3 = 3 (This checks out!)

• Equation 2: 3x - y = 10 3(3) - (-1) = 10 9 + 1 = 10 10 = 10 (This also checks out!)

Since our values for x and y satisfy both equations, we can confidently say that our solution is correct! This checking step is like the final proofread on an important document. It catches any errors and gives us the peace of mind that we've done the work accurately. So, never skip this step – it's a small investment of time that can save you from making mistakes.

Solution and Summary

So, after all that, what's our final answer? We've determined that the solution to the system of equations:

• 2x + 3y = 3
• 3x - y = 10

is x = 3 and y = -1. We often write this as an ordered pair: (3, -1). This ordered pair represents the point where the lines represented by these two equations intersect on a graph. It's the single point that satisfies both equations simultaneously.

Let's recap the steps we took to get here:

1. Isolate a variable: We chose the second equation and isolated y to get y = 3x - 10.
2. Substitute: We substituted 3x - 10 for y in the first equation, resulting in 2x + 3(3x - 10) = 3.
3. Solve for x: We simplified and solved the equation to find x = 3.
4. Substitute back: We plugged x = 3 back into y = 3x - 10 to find y = -1.
5. Check: We verified our solution by plugging x = 3 and y = -1 into both original equations.

By following these steps, you can confidently solve any system of equations using the substitution method! It's a powerful tool in your mathematical arsenal, and with practice, you'll become a master of it.

When to Use the Substitution Method

The substitution method is a fantastic tool, but it’s not always the best tool for every job. So, when should you reach for it? The substitution method shines in certain situations, and understanding these scenarios will help you become a more efficient problem-solver.

One of the key indicators that the substitution method might be a good choice is when one of the equations has a variable with a coefficient of 1 or -1. Remember, the first step in substitution is to isolate a variable, and this is much easier when the coefficient is already 1 or -1. In our example, the equation 3x - y = 10 was perfect for this because the y term had a coefficient of -1. This allowed us to isolate y without dealing with fractions, which can make the process much smoother.

Another situation where substitution excels is when one of the equations is already solved for one variable. Imagine you have a system where one equation is y = 2x + 5. In this case, the first step of substitution is already done for you! You can simply plug this expression for y into the other equation and solve for x. This saves you a step and makes the whole process quicker.

However, if both equations have coefficients other than 1 or -1 for all variables, and none of the equations are already solved for a variable, the elimination method might be a more efficient choice. The elimination method involves adding or subtracting the equations to eliminate one of the variables, and it can be particularly useful when the coefficients line up nicely for elimination. But don't worry, we'll explore the elimination method in another article!

Ultimately, the best method depends on the specific system of equations you're facing. But by understanding the strengths of the substitution method, you'll be well-equipped to tackle a wide range of problems.

Practice Makes Perfect: Try These Examples!

Okay, guys, now that we've gone through the theory and steps of the substitution method, it's time to put your knowledge to the test! The best way to truly master this technique is through practice. So, let's dive into a few more examples to solidify your understanding. Remember, the more you practice, the more comfortable and confident you'll become in solving systems of equations.

Here are a couple of systems you can try solving using the substitution method:

1. x + 2y = 7 3x - y = -1

2. 4x - 3y = 10 x + y = 1

For each system, follow the steps we discussed earlier:

• Isolate a variable in one equation.
• Substitute the expression into the other equation.
• Solve for the remaining variable.
• Substitute the value back to find the other variable.
• Check your solution.

Don't be afraid to make mistakes – they're a natural part of the learning process! If you get stuck, go back and review the steps we covered. Pay close attention to the substitution step and make sure you're distributing correctly. And remember, checking your solution is crucial to catch any errors.

As you work through these examples, you'll start to develop a feel for when the substitution method is the most efficient approach. You'll also get better at identifying the easiest variable to isolate and avoiding common pitfalls. So, grab your pencils, and let's get practicing!

Conclusion: Mastering the Substitution Method

Congratulations, you've made it to the end of our deep dive into the substitution method! You've learned a powerful technique for solving systems of equations, and you're well on your way to becoming a math whiz. Remember, the substitution method is just one tool in your problem-solving toolkit, but it's a valuable one to have.

We've covered the step-by-step process, from isolating a variable to checking your solution. We've discussed when the substitution method is the most effective choice, and we've even given you some practice problems to hone your skills. The key takeaway is that practice is essential. The more you use the substitution method, the more natural and intuitive it will become.

Solving systems of equations is a fundamental skill in mathematics, and it has applications in many different fields. Whether you're studying physics, economics, or computer science, you'll encounter situations where you need to solve for multiple variables simultaneously. By mastering methods like substitution, you're not just learning math; you're developing critical thinking and problem-solving abilities that will serve you well in all areas of life.

So, keep practicing, keep exploring, and never stop learning! And the next time you encounter a system of equations, remember the substitution method – it might just be the perfect solution.