Solving Systems of Equations: A Step-by-Step Guide

When it comes to algebra, systems of equations can seem like a maze—but guess what? With some guidance, you’re gonna navigate through it like a pro! Let’s break down how to find the values of x and y in the equations 2x + 2y = 14 and 3x - 2y = -4. Trust me; it’s easier than you think!

First off, we have our system of equations:

1. 2x + 2y = 14
2. 3x - 2y = -4

Now, we need to simplify things a bit. The first equation can be streamlined by dividing everything by 2. Why make it complicated? So, we get:

• x + y = 7

This new equation makes it clearer! From here, it’s easy-peasy to express y in terms of x. Rearranging gives us:

• y = 7 - x

Cool, right? Now let’s shift our focus to the second equation: 3x - 2y = -4. When you substitute our expression for y into this equation, it transforms the math puzzle into a one-step dance. Here’s what I mean:

We take:

• 3x - 2(7 - x) = -4

Now, let’s expand it:

• 3x - 14 + 2x = -4

Combine the like terms—don’t be shy about it:

• 5x - 14 = -4

Next, we need to get rid of that pesky -14 on the left side. So, we’ll add 14 to both sides:

• 5x = 10

Now, divide by 5 to find x:

• x = 2

Hang on a minute! We’re so close, but we still need to find y. Time to use our earlier expression for y. Plugging in x = 2 gives us:

• y = 7 - 2 = 5

Voilà! The values of x and y that satisfy the equations are x = 2 and y = 5.

You know what’s exciting? Solving equations like this isn't just about finding the right numbers; it's about building your problem-solving skills, which you’ll use in so many aspects of life. Whether it's figuring out a budget, optimizing resources, or even navigating the complexities of everyday decisions, the foundational skills you develop through algebra stick with you.

Before we wrap up, here’s a little tip. When tackling more complicated systems, don't hesitate to use graphing methods or matrices if they fit the bill. Sometimes jumping back and forth between methods gives us clearer insights. And remember, practice makes perfect! So don’t rush past solving practice problems. They’re your stepping stones to understanding.

In summary, while systems of equations can initially look daunting, tackling them becomes second nature with practice and the right techniques. Keep grinding, and every problem you solve will add to your confidence. Algebra is like any skill; the more you work at it, the better you get!