Solving Linear Equations: Find 4a - 2b!

Alright guys, let's dive into this math problem! We've got two lines, and we need to figure out where they cross each other. Once we know that point, we can calculate the value of a simple expression. Sounds fun, right?

Understanding the Problem

First, let's break down what we know. We have two equations:

1. 3x - 2y - 12 = 0
2. 5x - 2y = 3

The point where these lines intersect is (a, b). That means if we plug in 'a' for 'x' and 'b' for 'y' in both equations, they should both be true. Our mission is to find the value of 4a - 2b.

Why is this important? Well, solving systems of linear equations is a fundamental skill in math. It pops up everywhere – from physics and engineering to economics and computer science. Understanding how to solve these problems gives you a powerful tool for tackling real-world situations.

Solving the System of Equations

So, how do we find this magical point (a, b)? We've got a couple of options, but since the equations are nicely set up, let's use the elimination method. Notice that both equations have a '-2y' term. That's perfect for eliminating 'y'!

Step 1: Eliminate 'y'

Subtract the first equation from the second equation:

(5x - 2y) - (3x - 2y - 12) = 3 - 0

Simplify it:

5x - 2y - 3x + 2y + 12 = 3

2x + 12 = 3

Step 2: Solve for 'x'

Now we can easily solve for 'x':

2x = 3 - 12

2x = -9

x = -9/2

So, we've found that a = -9/2. Awesome!

Step 3: Solve for 'y'

Now that we know 'x', we can plug it back into either of the original equations to solve for 'y'. Let's use the second equation (it looks a bit simpler):

5x - 2y = 3

5*(-9/2) - 2y = 3

-45/2 - 2y = 3

-2y = 3 + 45/2

-2y = 6/2 + 45/2

-2y = 51/2

y = -51/4

Alright! We found that b = -51/4.

Calculating 4a - 2b

Now that we know a = -9/2 and b = -51/4, we can finally calculate 4a - 2b:

4a - 2b = 4*(-9/2) - 2*(-51/4)

= -36/2 + 102/4

= -18 + 51/2

= -36/2 + 51/2

= 15/2

Therefore, 4a - 2b = 15/2

Alternative Methods for Solving Linear Equations

While we used the elimination method here, it's worth knowing that there are other ways to crack these problems. Here are a couple:

• Substitution Method: Solve one equation for one variable (say, 'x' in terms of 'y'), and then substitute that expression into the other equation. This will give you an equation with only one variable, which you can solve. Then, plug that value back into either equation to find the other variable.
• Graphing: Graph both lines on a coordinate plane. The point where they intersect is your solution (a, b). This method is great for visualizing the problem, but it might not be the most accurate if the solution involves fractions or decimals.
• Matrices: For more complex systems of equations (with more than two variables), you can use matrices and techniques like Gaussian elimination or finding the inverse of a matrix. This is a more advanced topic, but it's super useful in many fields.

Why This Matters: Real-World Applications

You might be thinking, "Okay, I can solve these equations, but when will I ever use this in real life?" Well, here are a few examples:

• Mixing Problems: Imagine you're mixing two different solutions with different concentrations of a chemical. You can use a system of equations to figure out how much of each solution you need to get the desired concentration.
• Economics: Supply and demand curves can be represented as linear equations. The point where they intersect is the equilibrium point, which determines the price and quantity of goods in a market.
• Physics: Analyzing forces in equilibrium often involves solving systems of equations. For example, you might need to find the tension in different cables supporting a weight.
• Computer Graphics: Linear equations are used extensively in computer graphics to perform transformations like scaling, rotation, and translation of objects.

Tips and Tricks for Solving Linear Equations

• Check Your Work: Always plug your solution back into the original equations to make sure it works.
• Be Careful with Signs: A simple sign error can throw off your entire solution. Double-check your work, especially when dealing with negative numbers.
• Look for Easy Eliminations: If you see that the coefficients of one variable are the same (or easily made the same), elimination is usually the easiest method.
• Don't Be Afraid to Use Fractions: Sometimes the solution will involve fractions. Embrace them!
• Practice, Practice, Practice: The more you practice, the better you'll become at solving linear equations. Start with simple problems and gradually work your way up to more complex ones.

Conclusion

So, there you have it! We successfully found the value of 4a - 2b by solving a system of linear equations. Remember, this is a fundamental skill that will come in handy in many areas of your life. Keep practicing, and you'll become a master of linear equations in no time! And remember guys, math is not a monster. With the correct guide you can be a master.