Solving 4x + 2y = 18 And 5x + 3y = 24 Find X And Y
Hey guys! Today, we're diving into the exciting world of algebra to tackle a classic problem: solving a system of linear equations. Specifically, we're going to figure out the values of x and y that satisfy both of these equations:
- 4x + 2y = 18
- 5x + 3y = 24
Don't worry if this looks intimidating! We'll break it down step by step, using a method called elimination. By the end of this guide, you'll not only know the answer but also understand the process, so you can confidently solve similar problems in the future. Let's get started!
Understanding Systems of Equations
Before we jump into the solution, let's take a moment to understand what a system of equations actually represents. Think of each equation as a line drawn on a graph. The solution to the system is the point where these lines intersect. This point has an x-coordinate and a y-coordinate that make both equations true.
In our case, we have two equations, and therefore, two lines. Our goal is to find the one point (x, y) that lies on both lines simultaneously. There are several methods to solve systems of equations, including graphing, substitution, and elimination. We're focusing on elimination here because it's often the most efficient method for equations in this format.
Why Elimination?
The elimination method works by manipulating the equations so that when we add them together, one of the variables cancels out. This leaves us with a single equation in a single variable, which is much easier to solve. Once we find the value of one variable, we can plug it back into one of the original equations to find the value of the other variable. It's like a strategic game of simplification!
Getting Ready for Elimination
The key to successful elimination is to find a way to make the coefficients (the numbers in front of the variables) of either x or y opposites. For instance, if one equation has 2y and the other has -2y, adding the equations will eliminate y. If the coefficients aren't already opposites, we can multiply one or both equations by a constant to make them so.
Now, let's apply this to our specific problem. Looking at our equations:
- 4x + 2y = 18
- 5x + 3y = 24
We need to find a common multiple for either the coefficients of x (4 and 5) or the coefficients of y (2 and 3). The least common multiple of 2 and 3 is 6, which seems like the easier route. So, we'll aim to make the coefficients of y be 6 and -6.
Step-by-Step Solution using Elimination
Alright, let's walk through the elimination method step-by-step. This will help you grasp the process and apply it to other problems.
Step 1: Manipulate the Equations
To get the coefficients of y to be 6 and -6, we'll multiply the first equation by 3 and the second equation by -2:
- Equation 1 (multiplied by 3): 3 * (4x + 2y) = 3 * 18 => 12x + 6y = 54
- Equation 2 (multiplied by -2): -2 * (5x + 3y) = -2 * 24 => -10x - 6y = -48
Now our equations look like this:
- 12x + 6y = 54
- -10x - 6y = -48
Notice that the y terms have coefficients of 6 and -6. This is perfect!
Step 2: Add the Equations
Now, we add the two equations together. When we do this, the y terms will cancel out:
(12x + 6y) + (-10x - 6y) = 54 + (-48)
This simplifies to:
2x = 6
Step 3: Solve for x
Now we have a simple equation with just one variable, x. To solve for x, we divide both sides of the equation by 2:
2x / 2 = 6 / 2
So, x = 3
Great! We've found the value of x. Now we need to find the value of y.
Step 4: Substitute to Find y
To find y, we substitute the value of x (which is 3) into either of the original equations. Let's use the first original equation:
4x + 2y = 18
Substitute x = 3:
4 * 3 + 2y = 18
This simplifies to:
12 + 2y = 18
Step 5: Solve for y
Now we solve for y. First, subtract 12 from both sides:
2y = 18 - 12
2y = 6
Then, divide both sides by 2:
y = 6 / 2
So, y = 3
Step 6: State the Solution
We've done it! We've found the values of x and y that satisfy both equations. The solution is:
- x = 3
- y = 3
We can write this as an ordered pair: (3, 3). This represents the point where the two lines intersect on a graph.
Verification: Checking Our Solution
It's always a good idea to check our solution to make sure we haven't made any mistakes. We do this by plugging the values of x and y back into the original equations.
Checking Equation 1: 4x + 2y = 18
Substitute x = 3 and y = 3:
4 * 3 + 2 * 3 = 18
12 + 6 = 18
18 = 18 (This is true!)
Checking Equation 2: 5x + 3y = 24
Substitute x = 3 and y = 3:
5 * 3 + 3 * 3 = 24
15 + 9 = 24
24 = 24 (This is also true!)
Since our solution (3, 3) satisfies both original equations, we can be confident that it's correct. Awesome job!
Alternative Methods: Substitution
While we've focused on elimination, it's worth mentioning another common method for solving systems of equations: substitution. In substitution, you solve one equation for one variable and then substitute that expression into the other equation. This also results in a single equation with a single variable.
For example, in our first equation (4x + 2y = 18), we could solve for y:
2y = 18 - 4x
y = 9 - 2x
Then, we would substitute this expression for y into the second equation and solve for x. Once we find x, we can substitute it back into the equation y = 9 - 2x to find y.
Substitution can be particularly useful when one of the equations is already solved (or easily solved) for one variable. However, for this particular problem, elimination was arguably more straightforward due to the structure of the equations.
Real-World Applications of Systems of Equations
Systems of equations aren't just abstract math problems; they have tons of real-world applications. Here are a few examples:
- Mixture Problems: Imagine you're blending two types of coffee beans with different prices to create a specific blend. Systems of equations can help you determine how much of each type of bean you need.
- Distance-Rate-Time Problems: If you have two objects moving at different speeds, you can use systems of equations to figure out when and where they'll meet.
- Supply and Demand: Economists use systems of equations to model the relationship between the supply and demand of goods and services.
- Circuit Analysis: Electrical engineers use systems of equations to analyze the flow of current in electrical circuits.
- Curve Fitting: Scientists and engineers often use systems of equations to find the equation of a curve that best fits a set of data points.
These are just a few examples, but they illustrate how systems of equations are a powerful tool for solving problems in a variety of fields. Understanding how to solve them is a valuable skill!
Practice Makes Perfect: More Examples
The best way to master solving systems of equations is to practice! Let's look at a couple more examples (without going into quite as much detail) to solidify your understanding.
Example 2:
Solve the system:
- 2x - y = 5
- x + 3y = 6
Solution:
- Multiply the first equation by 3: 6x - 3y = 15
- Add the modified first equation to the second equation: 7x = 21
- Solve for x: x = 3
- Substitute x = 3 into the second equation: 3 + 3y = 6
- Solve for y: y = 1
- Solution: (x, y) = (3, 1)
Example 3:
Solve the system:
- 3x + 2y = 7
- 2x - 5y = -8
Solution:
- Multiply the first equation by 5: 15x + 10y = 35
- Multiply the second equation by 2: 4x - 10y = -16
- Add the modified equations: 19x = 19
- Solve for x: x = 1
- Substitute x = 1 into the first equation: 3 + 2y = 7
- Solve for y: y = 2
- Solution: (x, y) = (1, 2)
Keep practicing, and you'll become a pro at solving systems of equations in no time!
Tips and Tricks for Solving Systems of Equations
Here are a few extra tips and tricks that can make solving systems of equations even easier:
- Look for Easy Eliminations: Before diving into multiplication, check if you can eliminate a variable directly by adding or subtracting the equations as they are. Sometimes, the coefficients are already opposites or easily made opposites with a simple multiplication.
- Choose the Easiest Variable to Eliminate: When deciding which variable to eliminate, choose the one that requires the least amount of multiplication. This can save you time and reduce the chances of making a mistake.
- Be Careful with Signs: Pay close attention to the signs (positive and negative) when multiplying and adding equations. A small sign error can throw off your entire solution.
- Organize Your Work: Write neatly and keep your steps organized. This will help you track your progress and make it easier to spot any errors.
- Check Your Work: As we demonstrated earlier, always check your solution by substituting the values of x and y back into the original equations. This is the best way to ensure accuracy.
- Consider Special Cases: Be aware of special cases, such as systems with no solutions (parallel lines) or infinitely many solutions (the same line). If you end up with a contradiction (e.g., 0 = 1) when solving, the system has no solutions. If you end up with an identity (e.g., 0 = 0), the system has infinitely many solutions.
By keeping these tips in mind, you'll be well-equipped to tackle any system of equations that comes your way!
Conclusion: Mastering Systems of Equations
So, there you have it! We've walked through a comprehensive guide on solving the system of equations 4x + 2y = 18 and 5x + 3y = 24. We used the elimination method, found the solution (x = 3, y = 3), and even checked our answer. We also touched on the substitution method, real-world applications, and some helpful tips and tricks.
Remember, the key to mastering systems of equations is practice. The more problems you solve, the more comfortable you'll become with the process. Don't be afraid to make mistakes – they're a part of learning. Just keep practicing, and you'll be solving systems of equations like a pro in no time!
If you found this guide helpful, be sure to share it with your friends and classmates. And if you have any questions or would like to see more examples, let us know in the comments below. Keep up the great work, guys!
