Solving Equations: Find X In -9x + 4y = 8 And -3x - Y = 4
Hey guys! Today, we're diving into a classic algebra problem where we need to find the value of 'x' that satisfies two equations. This type of problem often pops up in math classes and standardized tests, so let's break it down step-by-step to make sure we understand exactly how to tackle it. Our mission, should we choose to accept it, is to find the value of x that works in both -9x + 4y = 8 and -3x - y = 4, assuming y is the same in both. Buckle up, because we're about to get our math on!
Understanding the Problem
Before we jump into solving, let's make sure we really grasp what the problem is asking. We've got two equations, each with two variables (x and y). Our goal isn't just to find any values for x and y, but specifically the value of x when y is consistent across both equations. This means the y value that makes the first equation true also has to make the second equation true. Think of it like finding the x coordinate where two lines intersect if you were to graph these equations. The key here is that we're dealing with a system of equations, and we need to find a solution that fits both.
Why This Matters
You might be thinking, "Okay, cool, but why do I need to know this?" Well, solving systems of equations is a fundamental skill in algebra and beyond. It shows up in all sorts of real-world applications, from engineering and physics to economics and computer science. For instance, you might use systems of equations to model the forces acting on a bridge, to calculate the optimal mix of investments in a portfolio, or to design algorithms for machine learning. So, mastering this skill isn't just about acing your math test; it's about building a foundation for tackling complex problems in various fields. Plus, the logical thinking and problem-solving strategies you develop by working through these types of problems are valuable in any area of life.
Setting Up for Success
Now that we understand the problem, let's talk strategy. There are a couple of common methods for solving systems of equations: substitution and elimination. For this particular problem, the elimination method might be a bit cleaner, but we'll walk through both approaches so you can see how they work and choose the one that clicks best for you. The main idea behind elimination is to manipulate the equations so that when you add them together, one of the variables cancels out, leaving you with a single equation in one variable. With substitution, you solve one equation for one variable and then substitute that expression into the other equation. No matter which method we choose, the goal is the same: to simplify the problem into something we can easily solve.
Solving the Equations Using Elimination
The elimination method is our first strategy for tackling this problem. The beauty of this method is that it allows us to strategically eliminate one variable, making it easier to solve for the other. Let's see how it works with our equations:
- -9x + 4y = 8
- -3x - y = 4
Step-by-Step Elimination
Our goal here is to make either the x or y coefficients opposites so that when we add the equations, one variable cancels out. Looking at the equations, the y terms seem like a good place to start because we have 4y in the first equation and -y in the second. If we multiply the second equation by 4, we'll get -4y, which is the opposite of 4y. Let's do it:
Multiply equation (2) by 4:
4 * (-3x - y) = 4 * 4
This simplifies to:
-12x - 4y = 16
Now we have a new system of equations:
- -9x + 4y = 8
- -12x - 4y = 16
See how the y terms are lined up and ready to cancel each other out? This is the magic of the elimination method! Next, we'll add the two equations together.
Adding the Equations
Now for the main event: adding the two equations together. We'll add the left sides and the right sides separately:
(-9x + 4y) + (-12x - 4y) = 8 + 16
Combine like terms:
-9x - 12x + 4y - 4y = 24
-21x = 24
Boom! The y terms canceled out, leaving us with a simple equation in terms of x. Now, we just need to solve for x.
Solving for x
We've got -21x = 24. To isolate x, we'll divide both sides of the equation by -21:
x = 24 / -21
Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 3:
x = -8 / 7
So, we've found that x = -8/7. That wasn't so bad, right? We used the elimination method to strategically cancel out the y variable and solve for x. But just to be thorough, let's see how we'd tackle this same problem using the substitution method.
Solving the Equations Using Substitution
Now, let's try a different approach: the substitution method. This technique involves solving one equation for one variable and then substituting that expression into the other equation. It's like a mathematical relay race, where we pass information from one equation to the next. Let's see how it works with our system:
- -9x + 4y = 8
- -3x - y = 4
Step-by-Step Substitution
The first step in the substitution method is to pick one equation and solve it for one of the variables. It doesn't matter which equation or which variable you choose, but it's often easiest to pick the one that looks simplest to isolate. In this case, the second equation, -3x - y = 4, looks like a good candidate because the coefficient of y is -1. Let's solve this equation for y.
Add 3x to both sides:
-y = 3x + 4
Multiply both sides by -1 to get y by itself:
y = -3x - 4
Now we have an expression for y in terms of x. This is the key piece of information we'll substitute into the other equation.
Substituting the Expression
Next, we'll substitute our expression for y (which is -3x - 4) into the first equation, -9x + 4y = 8. This means we'll replace the y in the first equation with -3x - 4:
-9x + 4(-3x - 4) = 8
Now we have an equation with just one variable, x. Let's simplify and solve it.
Solving for x
Distribute the 4:
-9x - 12x - 16 = 8
Combine like terms:
-21x - 16 = 8
Add 16 to both sides:
-21x = 24
Divide both sides by -21:
x = 24 / -21
Simplify the fraction:
x = -8 / 7
Hey, look at that! We got the same answer for x as we did using the elimination method: x = -8/7. This is a good sign that we're on the right track. Now, let's take a moment to reflect on what we've done and why these methods work.
Verifying the Solution and Finding y (Optional)
We've found that x = -8/7. To be absolutely sure we've got the right answer, and also to find the corresponding y value, we can plug our x value back into either of the original equations. This is a great way to double-check your work and make sure everything lines up.
Plugging x into Equation 2
Let's use the second equation, -3x - y = 4, because it looks a bit simpler. Substitute x = -8/7:
-3(-8/7) - y = 4
Simplify:
24/7 - y = 4
To get rid of the fraction, let's rewrite 4 as 28/7:
24/7 - y = 28/7
Subtract 24/7 from both sides:
-y = 4/7
Multiply both sides by -1:
y = -4/7
So, we've found that y = -4/7. Now, let's just make super sure by plugging both x and y into the first equation.
Plugging x and y into Equation 1
Our first equation is -9x + 4y = 8. Let's substitute x = -8/7 and y = -4/7:
-9(-8/7) + 4(-4/7) = 8
Simplify:
72/7 - 16/7 = 8
56/7 = 8
8 = 8
It checks out! Both equations are satisfied by x = -8/7 and y = -4/7. This gives us a lot of confidence that we've found the correct solution.
Wrapping It Up
Okay, guys, we did it! We successfully found the value of x that satisfies both equations -9x + 4y = 8 and -3x - y = 4. We walked through two different methods – elimination and substitution – and saw how each one can be used to solve the same problem. We even took the extra step to verify our solution and find the corresponding y value.
Key Takeaways
- Systems of equations are sets of two or more equations with the same variables, and solving them means finding values for the variables that make all the equations true.
- The elimination method involves manipulating the equations so that when you add them together, one variable cancels out.
- The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.
- Verifying your solution by plugging the values back into the original equations is always a good idea to catch any mistakes.
Final Thoughts
Solving systems of equations is a core skill in algebra, and it's something you'll use again and again in math and science. The key is to practice and become comfortable with the different methods. Don't be afraid to try both elimination and substitution to see which one you prefer for a particular problem. And remember, verifying your solution is always a smart move. Keep up the great work, and you'll be a system-solving pro in no time!