Solving System Of Equations 4x + 2y = 12 And -4x - 5y = 6
Hey guys! Today, we're going to dive into solving a system of linear equations. Specifically, we'll be tackling the equations 4x + 2y = 12 and -4x - 5y = 6. This kind of problem is super common in algebra, and once you get the hang of it, it's actually pretty fun. We'll break it down step by step so it’s crystal clear. So, let's jump right in!
Understanding Systems of Equations
Before we get into the nitty-gritty, let's quickly recap what a system of equations is. Essentially, it's a set of two or more equations that we're trying to solve simultaneously. This means we're looking for values of the variables (in this case, x and y) that satisfy all the equations in the system. Think of it like finding the sweet spot that makes everything balance out.
Why do we care about this? Well, systems of equations pop up everywhere in real life! From figuring out the right mix of ingredients in a recipe to modeling complex relationships in science and economics, these systems help us solve a wide range of problems. Mastering this skill is super valuable, so let’s get to it!
When tackling a system of equations, we're essentially trying to find the point where the lines represented by the equations intersect. Each linear equation, like the ones we're working with today, can be visualized as a straight line on a graph. The solution to the system is the coordinate point (x, y) where these lines cross each other. This point is the only solution that works for both equations. Sometimes the lines might be parallel (no solution) or they might be the exact same line (infinite solutions), but in our case, we're looking for that single, perfect intersection point.
Now, there are several methods we can use to solve systems of equations, such as substitution, elimination, and graphing. Each method has its own strengths, and the best one to use often depends on the specific equations you're dealing with. For this problem, we’re going to use the elimination method, which is particularly handy when the coefficients of one of the variables are opposites or easy to make opposites. You'll see why this method is so neat in just a moment!
The Elimination Method: A Step-by-Step Guide
Okay, let’s get our hands dirty with the elimination method. This method is all about strategically adding or subtracting the equations in our system to eliminate one of the variables. By doing this, we end up with a single equation in just one variable, which is much easier to solve. Once we find the value of that variable, we can plug it back into one of the original equations to find the value of the other variable. It’s like a clever little puzzle!
Here’s how it works for our system:
-
Write down the equations:
We've got:
- 4x + 2y = 12
- -4x - 5y = 6
-
Notice the magic:
Look closely, guys! Do you see anything special about the coefficients of x? That's right, we have a 4x in the first equation and a -4x in the second. These are opposites! This is perfect for the elimination method because when we add the equations together, the x terms will cancel each other out.
-
Add the equations:
Now, we add the left-hand sides and the right-hand sides separately:
(4x + 2y) + (-4x - 5y) = 12 + 6
Notice how the 4x and -4x neatly cancel out:
-3y = 18
Woo-hoo! We've eliminated x and have a simple equation in y.
-
Solve for y:
To isolate y, we divide both sides of the equation by -3:
y = 18 / -3
y = -6
Awesome! We've found the value of y.
-
Substitute y back in:
Now that we know y = -6, we can plug this value back into either of the original equations to solve for x. It doesn't matter which one we choose; we'll get the same answer. Let's use the first equation, 4x + 2y = 12, because it looks a bit simpler:
4x + 2(-6) = 12
4x - 12 = 12
-
Solve for x:
To isolate x, we first add 12 to both sides:
4x = 24
Then, we divide both sides by 4:
x = 24 / 4
x = 6
Fantastic! We've found the value of x.
-
Write the solution:
We've found that x = 6 and y = -6. So, the solution to the system of equations is the ordered pair (6, -6).
Checking Our Solution
Alright, we've got our solution (6, -6), but before we celebrate, it's always a good idea to check our work. This is a crucial step to make sure we haven't made any sneaky errors along the way. To check, we simply plug our values of x and y back into both of the original equations and see if they hold true. If they do, we can be confident in our answer. If not, we know we need to go back and hunt for any mistakes.
Let's do the check:
-
First equation: 4x + 2y = 12
Substitute x = 6 and y = -6:
4(6) + 2(-6) = 12
24 - 12 = 12
12 = 12
Yep, it checks out!
-
Second equation: -4x - 5y = 6
Substitute x = 6 and y = -6:
-4(6) - 5(-6) = 6
-24 + 30 = 6
6 = 6
It checks out here too!
Since our solution (6, -6) satisfies both equations, we can confidently say that it is the correct solution to the system. High five!
Why This Matters: Real-World Applications
Now that we've successfully solved this system of equations, you might be wondering,
