Solve Systems Of Equations: A Step-by-Step Guide
Hey guys! Today, we're diving deep into the awesome world of mathematics, specifically tackling a problem that might seem a little tricky at first glance: finding the solution to a system of equations. You know, those situations where you have two or more equations with the same variables, and you need to find the values that make all of them true at the same time. It's like a mathematical detective mission where you're hunting for the one perfect combination! We'll be working through a specific example, and by the end, you'll be a pro at spotting these solutions. So, grab your notebooks, get comfy, and let's unravel this mystery together!
Understanding the Problem: What Are We Even Looking For?
Alright, let's break down what it actually means to find the solution to a system of equations. Imagine you have two lines on a graph. Each line represents an equation. The solution to the system is the exact point (or points!) where those two lines intersect. Think about it: at that intersection point, both equations must be satisfied simultaneously. It's the only spot on the graph that belongs to both lines. Our specific problem gives us two equations:
y = 2x - 3.5x - 2y = -14
Our mission, should we choose to accept it, is to find the pair of (x, y) values that makes both of these statements true. We've been given some options: A. (-7, 3.5), B. (3.5, -7), C. (7, 10.5), and D. (10.5, 7). We need to figure out which of these coordinate pairs is the golden ticket! There are several ways to solve systems of equations, but for this particular problem, substitution seems like a super straightforward method, especially since the first equation is already solved for y. Let's get our detective hats on and start plugging in!
Method 1: The Substitution Superstar
The substitution method is a classic and, honestly, one of my favorite ways to solve systems of equations when one variable is already isolated. In our problem, the first equation, y = 2x - 3.5, is perfect for this. It tells us exactly what y is in terms of x. So, what do we do? We take this expression for y and substitute it into the second equation wherever we see y. It's like a clever mathematical swap! Let's see how this plays out. Our second equation is x - 2y = -14. We're going to replace that y with (2x - 3.5):
x - 2(2x - 3.5) = -14
Now, we've got a single equation with only one variable, x. This is fantastic because we know how to solve single-variable equations! First, we need to distribute that -2 to both terms inside the parentheses:
x - 4x + 7 = -14
Next, let's combine the like terms on the left side (the x terms):
-3x + 7 = -14
Our goal is to get x by itself. So, we'll subtract 7 from both sides of the equation:
-3x = -14 - 7
-3x = -21
Finally, to isolate x, we divide both sides by -3:
x = -21 / -3
x = 7
Boom! We've found the x-coordinate of our solution. Now, this is where it gets exciting. We know x = 7, but we need the full (x, y) pair. Remember, the solution must satisfy both original equations. We can use our found value of x and plug it back into either of the original equations to find y. The first equation, y = 2x - 3.5, looks like the easiest one to use.
Let's substitute x = 7 into it:
y = 2(7) - 3.5
y = 14 - 3.5
y = 10.5
And there we have it! Our solution is the coordinate pair (7, 10.5). This means that when x is 7 and y is 10.5, both of our original equations hold true. Pretty neat, right? This substitution method really breaks down complex problems into manageable steps. It's all about systematically replacing parts of an equation with equivalent expressions until you get to a solvable form. Don't shy away from it; it's a powerful tool in your math arsenal!
Method 2: The Elimination Ace
While substitution was super handy here, it's also good to know other methods. Let's try the elimination method on the same system of equations. This method is particularly useful when neither equation is conveniently solved for a variable, or when the variables are nicely aligned. Our equations are:
y = 2x - 3.5x - 2y = -14
To use elimination, we first want both equations in the standard form, Ax + By = C. So, let's rearrange the first equation:
Subtract 2x from both sides:
-2x + y = -3.5
Now our system looks like this:
-2x + y = -3.5x - 2y = -14
The goal of elimination is to manipulate one or both equations (by multiplying them by a constant) so that when you add or subtract the equations, one of the variables cancels out completely. We want to make the coefficients of either x or y opposites. Let's look at the y coefficients: we have +1y in the first equation and -2y in the second. If we multiply the first equation by 2, the y coefficient will become +2y, which is the opposite of -2y.
Multiply the first equation by 2:
2 * (-2x + y) = 2 * (-3.5)
-4x + 2y = -7
Now our modified system is:
-4x + 2y = -7x - 2y = -14
Look at that! The y terms (+2y and -2y) are opposites. If we add these two equations together, the y terms will cancel out:
(-4x + 2y) + (x - 2y) = -7 + (-14)
Combine like terms:
-4x + x + 2y - 2y = -7 - 14
-3x + 0 = -21
-3x = -21
And again, we solve for x by dividing both sides by -3:
x = -21 / -3
x = 7
Awesome! We got the same x value as with substitution. Now, just like before, we plug this value of x back into one of the original equations to find y. Let's use the first original equation, y = 2x - 3.5, because it's already set up to find y.
y = 2(7) - 3.5
y = 14 - 3.5
y = 10.5
So, using the elimination method, we also arrive at the solution (7, 10.5). It’s super reassuring when different methods give you the same answer. It means you're definitely on the right track! The elimination method is a beast when the numbers align perfectly for cancellation. It often saves you from dealing with fractions if you're careful with your multiplications.
Verification: Did We Nail It?
Okay, we've found our potential solution (7, 10.5) using two different methods. But in math, especially when you're solving problems like this, verification is key. It's your chance to double-check your work and be 100% sure you've got the right answer. How do we verify? We take our solution (x=7, y=10.5) and plug these values back into both of the original equations. If the equations hold true (meaning both sides of the equals sign are the same), then our solution is correct!
Let's test it with the first equation: y = 2x - 3.5
Substitute x = 7 and y = 10.5:
10.5 = 2(7) - 3.5
10.5 = 14 - 3.5
10.5 = 10.5
This equation is TRUE! Great start.
Now, let's test it with the second equation: x - 2y = -14
Substitute x = 7 and y = 10.5:
7 - 2(10.5) = -14
7 - 21 = -14
-14 = -14
This equation is ALSO TRUE! Since our solution (7, 10.5) satisfies both original equations, we can confidently say that this is indeed the solution to the system.
Looking back at the options provided:
- A.
(-7, 3.5) - B.
(3.5, -7) - C.
(7, 10.5) - D.
(10.5, 7)
Our verified solution, (7, 10.5), matches option C. So, option C is our correct answer, guys!
Why This Matters: Real-World Connections
So, why bother learning how to solve systems of equations? It might seem like just abstract math problems, but believe it or not, systems of equations pop up all over the place in the real world. Think about planning a budget: you might have two different spending goals (your equations) and need to figure out how much you can allocate to different categories (your variables) to meet both. In science, when you're dealing with different reactions or physical laws, you often need to solve systems of equations to find unknown quantities.
Even in everyday scenarios, like comparing phone plans or figuring out the best deal on two different services, you're essentially setting up and solving a system of equations. For example, one plan might have a lower monthly fee but higher per-minute charges, while another has a higher monthly fee but lower per-minute charges. You can write equations for the total cost of each plan based on minutes used, and then solve the system to find out at what point (or how many minutes) one plan becomes cheaper than the other. It's all about finding that sweet spot where conditions are met for multiple factors simultaneously. Mastering these algebraic techniques gives you powerful tools to analyze and solve problems in a logical, structured way, making better decisions whether you're in a classroom or navigating life's complexities. So, keep practicing, keep exploring, and remember that math is a language that helps us understand and shape the world around us!