Solving System Of Equations 3x+2y=15 And 4x+3y=12
Hey everyone! Today, we're diving into the fascinating world of solving system of equations. We've got a cool problem to tackle: 3x + 2y = 15 and 4x + 3y = 12. Now, this might seem daunting at first, but trust me, with the right approach, it's totally manageable. We'll break it down step by step, so even if you're just starting out with algebra, you'll be able to follow along. We will explore different methods to solve this system, providing a comprehensive understanding. Let's jump right in and make math a little less mysterious and a lot more fun!
Understanding the Problem
Before we jump into solving this system of equations, guys, it's super important to really understand what we're dealing with. When we talk about systems of equations, we're basically looking at two or more equations that share the same variables. In our case, we have two equations:
- 3x + 2y = 15
- 4x + 3y = 12
These equations both involve the variables 'x' and 'y'. Our mission, should we choose to accept it (and we do!), is to find the values of 'x' and 'y' that make both equations true at the same time. Think of it like finding the sweet spot where both equations agree. Graphically, each of these equations represents a line, and the solution to the system is the point where these lines intersect. This point of intersection gives us the 'x' and 'y' values that satisfy both equations. So, our goal is to find this magical point, and we've got a few awesome methods to get there.
Why is this important, you ask? Well, solving systems of equations isn't just a fun math puzzle (though it totally is that too!). It's a fundamental skill that pops up in all sorts of real-world situations. From calculating the break-even point in business to figuring out the optimal mix of ingredients in a recipe, systems of equations are everywhere. Mastering this skill opens doors to solving a wide range of problems, making it a super valuable tool in your mathematical toolkit. We're not just learning algebra here; we're building a foundation for tackling complex challenges in various fields. So, let's get ready to explore the methods that will help us crack this code and unlock the solutions!
Method 1: Elimination Method
The elimination method is a powerful technique for solving systems of equations, and it's one of my personal favorites because it's so systematic. The core idea behind the elimination method is to manipulate the equations in such a way that when we add or subtract them, one of the variables disappears. This leaves us with a single equation with a single variable, which is much easier to solve. Let's see how it works with our system:
- 3x + 2y = 15
- 4x + 3y = 12
Our first step is to find a way to make either the 'x' coefficients or the 'y' coefficients the same (but with opposite signs) in both equations. This way, when we add the equations, that variable will be eliminated. In this case, let's focus on eliminating 'x'. To do this, we can multiply the first equation by 4 and the second equation by -3. This will give us 12x in the first equation and -12x in the second equation.
Multiplying the first equation by 4, we get:
4 * (3x + 2y) = 4 * 15
12x + 8y = 60
Now, multiplying the second equation by -3, we have:
-3 * (4x + 3y) = -3 * 12
-12x - 9y = -36
Now we have our modified equations:
- 12x + 8y = 60
- -12x - 9y = -36
Here comes the magic! If we add these two equations together, the 'x' terms will cancel out:
(12x + 8y) + (-12x - 9y) = 60 + (-36)
This simplifies to:
-y = 24
Now, we can easily solve for 'y':
y = -24
Awesome! We've found the value of 'y'. But we're not done yet; we still need to find 'x'. To do this, we can substitute the value of 'y' back into either of our original equations. Let's use the first equation:
3x + 2y = 15
Substitute y = -24:
3x + 2*(-24) = 15
3x - 48 = 15
Now, solve for 'x':
3x = 63
x = 21
Ta-da! We've found both 'x' and 'y'. So, the solution to our system of equations is x = 21 and y = -24. We've successfully eliminated a variable and solved for the other, making this method a super handy tool in our problem-solving arsenal. This method showcases the power of algebraic manipulation to simplify complex problems. By strategically multiplying and adding equations, we transformed a tricky system into a straightforward solution.
Method 2: Substitution Method
Alright, let's switch gears and explore another fantastic method for solving systems of equations: the substitution method. This method is all about expressing one variable in terms of the other and then substituting that expression into the other equation. It's like a clever way of sneaking one equation's information into another to simplify things. Let's dive into how it works with our system:
- 3x + 2y = 15
- 4x + 3y = 12
The first step in the substitution method is to pick one of the equations and solve for one of the variables. It doesn't matter which one you choose, but sometimes one equation or variable might be easier to work with than others. In this case, let's choose the first equation and solve for 'x'.
Starting with the first equation:
3x + 2y = 15
Subtract 2y from both sides:
3x = 15 - 2y
Now, divide by 3 to isolate 'x':
x = (15 - 2y) / 3
Great! We've expressed 'x' in terms of 'y'. Now comes the fun part – substitution. We're going to take this expression for 'x' and substitute it into the second equation:
4x + 3y = 12
Substitute x = (15 - 2y) / 3:
4*((15 - 2y) / 3) + 3y = 12
Now we have an equation with only 'y', which we can solve. Let's simplify it:
(60 - 8y) / 3 + 3y = 12
Multiply the entire equation by 3 to get rid of the fraction:
60 - 8y + 9y = 36
Combine like terms:
y + 60 = 36
Subtract 60 from both sides:
y = -24
Awesome! We've found the value of 'y'. Now we just need to find 'x'. Remember that expression we found earlier for 'x' in terms of 'y'? Let's use that:
x = (15 - 2y) / 3
Substitute y = -24:
x = (15 - 2*(-24)) / 3
x = (15 + 48) / 3
x = 63 / 3
x = 21
Fantastic! We've found the value of 'x'. So, the solution to our system of equations using the substitution method is x = 21 and y = -24. We successfully isolated one variable, substituted its expression into the other equation, and solved for both variables. The substitution method really shines when one of the equations can be easily solved for one variable. It's a versatile tool that can make tackling systems of equations much more manageable.
Method 3: Graphical Method
Let's switch gears and visualize our equations using the graphical method! This method brings a visual element to solving systems of equations, allowing us to see the solution as the intersection point of two lines. It's like turning algebra into a picture, which can be super helpful for understanding what's going on. So, grab your graph paper (or your favorite graphing tool), and let's get visual!
We're working with the same system of equations:
- 3x + 2y = 15
- 4x + 3y = 12
To use the graphical method, we need to plot each equation as a line on a coordinate plane. To do this, it's often easiest to rewrite each equation in slope-intercept form (y = mx + b), where 'm' is the slope and 'b' is the y-intercept. Let's start with the first equation:
3x + 2y = 15
Subtract 3x from both sides:
2y = -3x + 15
Divide by 2:
y = (-3/2)x + (15/2)
So, the first equation has a slope of -3/2 and a y-intercept of 15/2 (or 7.5). Now, let's do the same for the second equation:
4x + 3y = 12
Subtract 4x from both sides:
3y = -4x + 12
Divide by 3:
y = (-4/3)x + 4
This equation has a slope of -4/3 and a y-intercept of 4.
Now that we have both equations in slope-intercept form, we can plot them on a graph. For each equation, we can start by plotting the y-intercept and then use the slope to find another point on the line. For example, for the first equation, we plot the point (0, 7.5). Since the slope is -3/2, we can move 2 units to the right and 3 units down to find another point. We can do the same for the second equation, plotting the y-intercept (0, 4) and using the slope -4/3 to find another point.
Once we have two points for each line, we can draw a line through them. The point where the two lines intersect is the solution to the system of equations. If we graph these two lines accurately, we'll see that they intersect at the point (21, -24). This means that x = 21 and y = -24, which matches the solution we found using the elimination and substitution methods.
The graphical method provides a really intuitive way to understand systems of equations. It allows us to see the solution as a visual representation, which can be especially helpful for those who are visual learners. It also highlights the connection between algebra and geometry. However, it's worth noting that the graphical method might not always be the most precise, especially if the solution involves fractions or decimals that are hard to read accurately from a graph. In such cases, the algebraic methods like elimination and substitution might be more reliable. But as a way to visualize and understand the concept, the graphical method is a fantastic tool to have in your mathematical toolkit.
Conclusion
Woo-hoo! We've journeyed through the world of solving systems of equations and explored not one, not two, but three different methods to tackle our problem: 3x + 2y = 15 and 4x + 3y = 12. We started with the elimination method, where we strategically manipulated the equations to eliminate one variable and solve for the other. Then, we dived into the substitution method, where we expressed one variable in terms of the other and substituted that expression into the other equation. Finally, we put on our visual hats and used the graphical method to plot the equations as lines and find the solution at their intersection point.
Through this exploration, we've discovered that the solution to our system of equations is x = 21 and y = -24. And the awesome part is that we arrived at this solution using three distinct approaches, which reinforces the validity of our answer. Each method brought its own unique perspective to the problem, highlighting the versatility of mathematical tools. The elimination method is great for its systematic approach, the substitution method shines when one variable can be easily isolated, and the graphical method offers a visual understanding of the solution.
Mastering these methods is like adding powerful tools to your mathematical toolbox. It's not just about finding the right answer; it's about developing problem-solving skills that can be applied in various contexts. Understanding systems of equations opens doors to tackling real-world problems in fields like science, engineering, economics, and even everyday decision-making. So, keep practicing, keep exploring, and remember that each method has its strengths and can be the perfect fit for different situations. You've got this! Now go out there and conquer those equations!
