Solving Systems Of Equations 3x+3y=9 And 8x+12y=20 A Comprehensive Guide

Aug 6, 2025 by ADMIN 73 views

Solving the System of Equations: A Comprehensive Guide

Hey guys! Today, we're diving into the exciting world of solving systems of equations. Don't worry if it sounds intimidating – we'll break it down step by step, making it super easy to understand. We'll tackle a specific system as an example, but the techniques we learn can be applied to many different problems. So, let's get started and become equation-solving masters!

The System We're Tackling

Our mission, should we choose to accept it (and we do!), is to solve the following system of equations:

3x + 3y = 9
8x + 12y = 20

This might look like a jumble of numbers and letters at first, but fear not! It's simply a set of two equations, each with two unknown variables, x and y. Our goal is to find the values of x and y that satisfy both equations simultaneously. Think of it like finding the perfect combination that unlocks a secret code.

Why Solve Systems of Equations?

You might be wondering, "Why bother learning this stuff?" Well, solving systems of equations is a fundamental skill in mathematics and has applications in various real-world scenarios. From calculating the optimal mix of ingredients in a recipe to determining the trajectory of a rocket, these systems help us model and solve problems in science, engineering, economics, and many other fields. So, mastering this skill is definitely worth your while!

Methods to the Madness: Different Approaches

There are several ways to crack this equation code, and we'll explore a couple of popular methods: substitution and elimination. Each method has its strengths, and choosing the right one can make the solving process much smoother. Think of them as different tools in your mathematical toolbox – the more tools you have, the better equipped you are to tackle any problem.

Method 1: Substitution – The Art of Replacement

The substitution method is like a clever game of replacing one thing with another. The basic idea is to solve one equation for one variable (let's say, x) and then substitute that expression into the other equation. This eliminates one variable, leaving us with a single equation in one variable, which we can easily solve. Let's see how it works with our system:

Choose an equation and solve for one variable: Let's take the first equation, 3x + 3y = 9. We can solve for x as follows:
```
3x = 9 - 3y
x = (9 - 3y) / 3
x = 3 - y
```
Now we have an expression for x in terms of y. It's like we've found a secret identity for x!
Substitute the expression into the other equation: Next, we'll substitute this expression for x (3 - y) into the second equation, 8x + 12y = 20:
```
8(3 - y) + 12y = 20
```
Notice that we've replaced x with its equivalent expression, leaving us with an equation containing only y. We've successfully eliminated x!
Solve for the remaining variable: Now we have a simple equation in y, which we can solve using basic algebra:
```
24 - 8y + 12y = 20
4y = -4
y = -1
```
We've found the value of y! It's like discovering the first piece of the puzzle.
Substitute back to find the other variable: Now that we know y = -1, we can substitute this value back into either of the original equations (or the expression we found for x) to find x. Let's use the expression x = 3 - y:
```
x = 3 - (-1)
x = 3 + 1
x = 4
```
We've found the value of x! We've completed the puzzle!
Check your solution: It's always a good idea to check your solution by plugging the values of x and y back into both original equations to make sure they hold true. Let's do that:
- Equation 1: 3x + 3y = 9 3(4) + 3(-1) = 12 - 3 = 9 (Correct!)
- Equation 2: 8x + 12y = 20 8(4) + 12(-1) = 32 - 12 = 20 (Correct!)
Our solution checks out! We've successfully solved the system using substitution.

Method 2: Elimination – The Art of Cancellation

The elimination method, also known as the addition method, takes a different approach. The core idea is to manipulate the equations so that the coefficients of one of the variables are opposites (e.g., 3 and -3). Then, we can add the equations together, and that variable will be eliminated, leaving us with a single equation in one variable. Let's see how this works with our system:

Multiply equations to create opposite coefficients: We want to make the coefficients of either x or y opposites. Let's target x. To do this, we can multiply the first equation by -8 and the second equation by 3:
- Equation 1 (multiplied by -8): -24x - 24y = -72
- Equation 2 (multiplied by 3): 24x + 36y = 60
Notice that the coefficients of x are now -24 and 24, which are opposites. We've set the stage for elimination!
Add the equations together: Now, we add the two modified equations together:
```
(-24x - 24y) + (24x + 36y) = -72 + 60
```
The x terms cancel out, leaving us with:
```
12y = -12
```
We've successfully eliminated x!
Solve for the remaining variable: We now have a simple equation in y, which we can solve:
```
y = -1
```
We've found the value of y! Just like with substitution, we've got the first piece of the puzzle.
Substitute back to find the other variable: We substitute y = -1 back into either of the original equations to find x. Let's use the first equation, 3x + 3y = 9:
```
3x + 3(-1) = 9
3x - 3 = 9
3x = 12
x = 4
```
We've found the value of x! The puzzle is complete!
Check your solution: Again, let's check our solution by plugging the values of x and y back into both original equations:
- Equation 1: 3x + 3y = 9 3(4) + 3(-1) = 12 - 3 = 9 (Correct!)
- Equation 2: 8x + 12y = 20 8(4) + 12(-1) = 32 - 12 = 20 (Correct!)
Our solution is verified! We've conquered the system using elimination.

The Grand Finale: The Solution

Both the substitution and elimination methods led us to the same solution: x = 4 and y = -1. This means that the point (4, -1) is the intersection point of the two lines represented by the equations. It's the one and only solution that satisfies both equations simultaneously.

Choosing Your Weapon: Which Method is Best?

So, which method should you use? Well, it often depends on the specific system of equations you're dealing with.

Substitution is a good choice when one of the equations is already solved for one variable or can be easily solved. It's like finding a loose thread and pulling on it to unravel the whole problem.
Elimination shines when the coefficients of one of the variables are already opposites or can be easily made opposites by multiplying the equations. It's like strategically lining up the pieces to make them cancel out.

In the end, the best method is the one you feel most comfortable with and that leads you to the solution most efficiently. Practice with both methods, and you'll develop a knack for choosing the best approach.

Level Up Your Skills: Practice Makes Perfect

Like any skill, solving systems of equations gets easier with practice. So, don't be afraid to tackle more problems! You can find tons of examples online, in textbooks, or even create your own systems to solve. The more you practice, the more confident you'll become in your equation-solving abilities.

Conclusion: You're an Equation-Solving Rockstar!

Congratulations! You've successfully navigated the world of solving systems of equations. We've explored two powerful methods, substitution and elimination, and applied them to a specific example. Remember, the key is to understand the underlying concepts and practice regularly. With these skills in your mathematical arsenal, you'll be well-equipped to tackle a wide range of problems. Keep up the great work, and happy equation-solving!