Substitution Method: Solving Systems Of Equations

Hey guys! Ever stared at a system of equations and thought, "How on earth do I solve this?" Well, today we're diving deep into the substitution method, a super handy technique to crack these algebraic puzzles. We'll also make sure our answers are in the cleanest form possible – reduced fractions. So, grab your notebooks, and let's get cracking on solving the system: $\left\{\begin{array}{l} -2 x+2 y=-10 \ 2 x+y=4 \end{array}\right.$

Understanding the Substitution Method

The substitution method is all about isolating one variable in one equation and then plugging that expression into the other equation. Think of it like a secret code: you're finding a piece of information (an expression for a variable) from one place and using it to unlock the solution in another. It's a systematic approach that guarantees you'll find the solution if one exists. The beauty of this method is its versatility; it works for any system of linear equations, especially when one of the variables already has a coefficient of 1 or -1, making isolation a breeze. We're aiming to reduce a system of two equations with two variables into a single equation with just one variable. Once we solve for that single variable, we can easily backtrack and find the value of the other. This process breaks down complex problems into manageable steps, making algebra less intimidating and more like solving a fun riddle. We'll be focusing on ensuring our final answers are in reduced fractions, meaning the numerator and denominator have no common factors other than 1. This is crucial for presenting your solutions accurately and professionally, especially in more advanced math contexts where simplified forms are often required. So, stick around, and by the end of this, you'll be a substitution pro, whipping out those reduced fractions like a seasoned mathematician!

Step-by-Step Solution

Alright team, let's break down our system:

1. Equation 1: $-2x + 2y = -10$
2. Equation 2: $2x + y = 4$

Our first mission is to isolate a variable. Looking at these equations, Equation 2 seems like the easiest place to start because the 'y' term has a coefficient of 1. This means we can get 'y' all by itself with minimal fuss.

From Equation 2, let's solve for 'y':

$y = 4 - 2x$

Now that we have an expression for 'y' (it's equal to $4 - 2x$), we're going to substitute this expression into the other equation (Equation 1). This is the core of the substitution method, guys!

Substitute $y = 4 - 2x$ into Equation 1:

$-2x + 2(4 - 2x) = -10$

See what we did there? We replaced every 'y' in Equation 1 with our handy expression $(4 - 2x)$. Now, this equation only has 'x' in it, which means we can solve for 'x'. Let's simplify and solve:

$-2x + 8 - 4x = -10$

Combine the 'x' terms:

$-6x + 8 = -10$

Now, we want to get the '-6x' term by itself. Subtract 8 from both sides:

$-6x = -10 - 8$

$-6x = -18$

Finally, divide both sides by -6 to find 'x':

$x = \frac{-18}{-6}$

$x = 3$

Awesome! We found our first value. Now, we need to find the value of 'y'. We can use the expression we found earlier, $y = 4 - 2x$, and plug in the value of 'x' we just discovered ($x=3$).

$y = 4 - 2(3)$

$y = 4 - 6$

$y = -2$

So, our solution appears to be $x = 3$ and $y = -2$.

Verification

Before we declare victory, it's always a good idea to check our solution by plugging these values back into both original equations. This ensures we haven't made any silly mistakes along the way.

Let's check Equation 1: $-2x + 2y = -10$

Substitute $x=3$ and $y=-2$:

$-2(3) + 2(-2) = -10$

$-6 - 4 = -10$

$-10 = -10$

Nailed it! Equation 1 checks out.

Now, let's check Equation 2: $2x + y = 4$

Substitute $x=3$ and $y=-2$:

$2(3) + (-2) = 4$

$6 - 2 = 4$

$4 = 4$

Bingo! Equation 2 also checks out. This confirms that our solution $(x, y) = (3, -2)$ is correct.

Reduced Fractions

Now, let's talk about reduced fractions. In this particular problem, our solutions for x and y turned out to be integers ($3$ and $-2$). Integers can be thought of as fractions with a denominator of 1 (e.g., $3 = \frac{3}{1}$, $-2 = \frac{-2}{1}$). Since the denominators are already 1, these are inherently in their simplest or reduced form. There are no common factors between the numerator and the denominator (other than 1) that could be cancelled out.

It's important to remember this requirement, especially if your calculations lead to fractions like $\frac{4}{8}$ or $\frac{6}{9}$. In those cases, you'd need to simplify them to their reduced forms, which would be $\frac{1}{2}$ and $\frac{2}{3}$ respectively, by dividing both the numerator and the denominator by their greatest common divisor. For our solution $(3, -2)$, since they are integers, they are already considered reduced fractions.

Conclusion

So there you have it, guys! We successfully tackled a system of equations using the substitution method. We isolated a variable, substituted it into the other equation, solved for one variable, and then used that result to find the other. Finally, we verified our solution and confirmed that our integer answers were indeed in their reduced fraction form. The substitution method is a fundamental skill in algebra, and practicing it will make solving more complex systems much easier down the line. Keep practicing, and you'll be an algebra whiz in no time!