Solving 3x+2y=11 And 2x-3y=3 A Step By Step Guide

Hey guys! Let's dive into the world of simultaneous equations! If you've ever stumbled upon a math problem that looks like a tangled mess of variables and numbers, you're in the right place. Today, we're going to unravel a classic example: the system of equations 3x + 2y = 11 and 2x - 3y = 3. We'll break down the steps, explain the concepts, and make sure you're a pro at solving these types of problems. So, grab your pencils, and let's get started!

Understanding Simultaneous Equations

First things first, what exactly are simultaneous equations? Think of them as a puzzle where you have two or more equations with two or more variables. The goal? To find the values of those variables that satisfy all equations at the same time. These equations pop up everywhere, from calculating the right mix of ingredients in a recipe to predicting the trajectory of a rocket. Knowing how to solve them is a super useful skill!

In our case, we have two equations and two variables (x and y), which is a common setup. There are a few methods we can use, but we'll focus on the substitution and elimination methods. These are the go-to techniques for tackling these problems, and once you master them, you'll feel like a math wizard.

Why Solve Simultaneous Equations?

Okay, so why bother learning this stuff? Well, simultaneous equations are more than just abstract math problems. They show up in real-world scenarios all the time. Imagine you're trying to figure out the cost of two different items based on a combined price, or maybe you're trying to balance a chemical equation. These are the kinds of situations where knowing how to solve simultaneous equations can be a lifesaver. Plus, it's a great way to sharpen your problem-solving skills, which is always a win!

Method 1 Substitution Method

The substitution method is all about isolating one variable in one equation and then plugging that expression into the other equation. This might sound a bit confusing at first, but trust me, it's easier than it seems. Let's walk through it step by step with our example equations:

1. 3x + 2y = 11
2. 2x - 3y = 3

Step 1 Isolate a Variable

We need to pick one equation and one variable to isolate. It's often easiest to choose a variable that has a coefficient of 1 (or -1) if there is one, as it simplifies the algebra. In our case, neither x nor y has a coefficient of 1 in either equation. So, let's go with the first equation, 3x + 2y = 11, and solve for y. We could also solve for x, but for this example, let's stick with y.

To isolate y, we'll first subtract 3x from both sides:

2y = 11 - 3x

Then, we'll divide both sides by 2:

y = (11 - 3x) / 2

Great! We've now expressed y in terms of x. This is a crucial step in the substitution method. We now have a way to replace y in the second equation with an expression involving x.

Step 2 Substitute the Expression

Now comes the fun part – substitution! We'll take the expression we just found for y, which is y = (11 - 3x) / 2, and plug it into the second equation, 2x - 3y = 3. This will give us an equation with only one variable, x, which we can solve.

So, we replace y in the second equation:

2x - 3((11 - 3x) / 2) = 3

See what we did there? We swapped y for the expression (11 - 3x) / 2. Now we have an equation that we can solve for x. But first, let's simplify it a bit.

Step 3 Simplify and Solve for x

To simplify, we'll start by multiplying both sides of the equation by 2 to get rid of the fraction:

2 * [2x - 3((11 - 3x) / 2)] = 2 * 3

This gives us:

4x - 3(11 - 3x) = 6

Now, distribute the -3:

4x - 33 + 9x = 6

Combine like terms:

13x - 33 = 6

Add 33 to both sides:

13x = 39

Finally, divide by 13:

x = 3

Awesome! We've found the value of x. Now we know that x = 3. But we're not done yet – we still need to find y.

Step 4 Solve for y

Now that we know x = 3, we can plug this value back into either of our original equations to solve for y. Or, even easier, we can use the expression we found for y in terms of x: y = (11 - 3x) / 2. Let's use that one.

Substitute x = 3:

y = (11 - 3 * 3) / 2

Simplify:

y = (11 - 9) / 2

y = 2 / 2

y = 1

There we go! We've found that y = 1.

Step 5 Check Your Solution

It's always a good idea to check your solution to make sure you didn't make any mistakes. Plug the values of x and y back into the original equations and see if they hold true.

For the first equation, 3x + 2y = 11:

3 * 3 + 2 * 1 = 9 + 2 = 11 (Correct!)

For the second equation, 2x - 3y = 3:

2 * 3 - 3 * 1 = 6 - 3 = 3 (Correct!)

Both equations are satisfied, so we know our solution is correct. The solution to the system of equations is x = 3 and y = 1.

Method 2: Elimination Method

Now, let's tackle the same system of equations using the elimination method. This method involves manipulating the equations so that when you add or subtract them, one of the variables cancels out. It's like magic, but it's just math!

Here are our equations again:

1. 3x + 2y = 11
2. 2x - 3y = 3

Step 1: Prepare for Elimination

To eliminate a variable, we need to make the coefficients of either x or y the same (but with opposite signs) in both equations. This way, when we add the equations, that variable will disappear. Looking at our equations, it might be easiest to eliminate y. To do this, we'll multiply the first equation by 3 and the second equation by 2. This will give us coefficients of 6y and -6y, which will cancel each other out when we add the equations.

Multiply the first equation by 3:

3 * (3x + 2y) = 3 * 11

9x + 6y = 33

Multiply the second equation by 2:

2 * (2x - 3y) = 2 * 3

4x - 6y = 6

Now we have a new system of equations:

1. 9x + 6y = 33
2. 4x - 6y = 6

Step 2: Eliminate a Variable

Now, we simply add the two equations together. Notice that the +6y and -6y will cancel out:

(9x + 6y) + (4x - 6y) = 33 + 6

This simplifies to:

13x = 39

Hey, this looks familiar! We're on the right track.

Step 3: Solve for x

Now, we solve for x by dividing both sides by 13:

x = 39 / 13

x = 3

Just like with the substitution method, we've found that x = 3.

Step 4: Solve for y

To find y, we can plug x = 3 back into either of our original equations. Let's use the first equation, 3x + 2y = 11:

3 * 3 + 2y = 11

9 + 2y = 11

Subtract 9 from both sides:

2y = 2

Divide by 2:

y = 1

We got y = 1 again! This is a good sign – it means we're consistent with our previous solution.

Step 5: Check Your Solution

Just like before, let's check our solution by plugging x = 3 and y = 1 back into the original equations:

For the first equation, 3x + 2y = 11:

3 * 3 + 2 * 1 = 9 + 2 = 11 (Correct!)

For the second equation, 2x - 3y = 3:

2 * 3 - 3 * 1 = 6 - 3 = 3 (Correct!)

Our solution checks out. So, using the elimination method, we've confirmed that x = 3 and y = 1 is the correct solution.

Choosing the Right Method

So, we've seen two methods for solving simultaneous equations: substitution and elimination. Which one should you use? Well, it depends on the problem. Here are a few guidelines:

• Substitution Method: This method is often best when one of the equations has a variable with a coefficient of 1 (or -1). It makes isolating that variable and substituting easier. If one equation is already solved for one variable (like y = something), substitution is usually the way to go.
• Elimination Method: This method is great when the coefficients of one of the variables are easily made the same (or opposite) by multiplying the equations. If the equations are in standard form (Ax + By = C), elimination can be more straightforward.

In our example, both methods worked well. But sometimes, one method will be significantly easier than the other. It's good to be comfortable with both so you can choose the best approach for each problem.

Common Mistakes to Avoid

Solving simultaneous equations can be tricky, and it's easy to make mistakes. Here are a few common pitfalls to watch out for:

• Sign Errors: Be super careful with negative signs. A small sign error can throw off your entire solution.
• Incorrect Distribution: When multiplying an equation, make sure you distribute the multiplier to all terms.
• Arithmetic Errors: Double-check your arithmetic, especially when adding, subtracting, multiplying, and dividing.
• Forgetting to Solve for Both Variables: Remember, you need to find the values of both x and y (or however many variables you have). Don't stop after solving for just one!
• Not Checking Your Solution: Always, always, always check your solution by plugging the values back into the original equations. This is the best way to catch mistakes.

Real-World Applications

We've talked about the mechanics of solving simultaneous equations, but let's take a moment to appreciate how useful this skill is in the real world. Here are just a few examples:

• Business and Economics: Businesses use simultaneous equations to model supply and demand, calculate costs and profits, and optimize resource allocation.
• Science and Engineering: Scientists and engineers use simultaneous equations to model physical systems, analyze circuits, solve for forces and stresses, and much more.
• Chemistry: Balancing chemical equations often involves solving simultaneous equations.
• Computer Graphics: Simultaneous equations are used to perform transformations in 3D graphics, like rotations and scaling.
• Everyday Life: Even in everyday situations, you might use simultaneous equations without realizing it. For example, if you're buying a combination of items with a set budget, you're essentially solving a system of equations.

Conclusion

So, there you have it! We've explored simultaneous equations, learned two powerful methods for solving them (substitution and elimination), and discussed how they apply to the real world. Solving 3x + 2y = 11 and 2x - 3y = 3 was just the beginning. With practice, you'll become a simultaneous equation-solving master!

Remember, the key is to break down the problem into manageable steps, stay organized, and double-check your work. And don't be afraid to ask for help if you get stuck. Math is a journey, and we're all in this together. Keep practicing, keep learning, and you'll be amazed at what you can achieve. Keep rocking, guys!