Solving 2x + 3y = 16 And 4x - 3y = 8 A Step-by-Step Guide

Jul 29, 2025 by ADMIN 58 views

Solving the System of Equations: 2x + 3y = 16 and 4x - 3y = 8

Hey guys! Let's dive into solving this system of equations. We've got two equations here: 2x + 3y = 16 and 4x - 3y = 8. Our goal is to find the values of x and y that satisfy both equations simultaneously. This is a classic problem in algebra, and there are several ways we can tackle it. We’ll explore the elimination method, which is particularly handy in this case because the coefficients of y are opposites. Stick around, and we'll break it down step by step so you can master this type of problem.

Understanding Systems of Equations

Before we jump into the solution, let's quickly recap what a system of equations is. A system of equations is simply a set of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that make all the equations true. In our case, we have two equations with two variables (x and y), and we need to find the pair of values (x, y) that fits both equations.

Systems of equations pop up all over the place in math and real-world applications. From balancing chemical equations to modeling supply and demand in economics, understanding how to solve them is a crucial skill. There are several methods to solve systems of equations, including substitution, elimination, and graphing. Each method has its strengths, and the best one to use often depends on the specific equations you're dealing with. For this problem, the elimination method is the most straightforward because the y terms have opposite coefficients.

The elimination method works by adding or subtracting the equations in a way that eliminates one of the variables. This leaves us with a single equation in one variable, which we can easily solve. Once we find the value of that variable, we can substitute it back into one of the original equations to find the value of the other variable. This might sound a bit abstract now, but don't worry—we'll see how it works in practice in just a moment. Remember, the key is to manipulate the equations so that either the x terms or the y terms cancel out when you add or subtract them. In our case, the y terms are perfectly set up for elimination because they have opposite signs. Let's get started with the solution!

The Elimination Method: A Step-by-Step Solution

The elimination method is our go-to strategy here because we notice something special about our equations: the y terms have opposite coefficients. This means if we add the equations together, the y terms will conveniently cancel each other out, leaving us with an equation involving only x. Let's walk through the steps:

Write down the equations:
- 2x + 3y = 16
- 4x - 3y = 8
Add the equations together: When we add the left-hand sides, we get (2x + 3y) + (4x - 3y). When we add the right-hand sides, we get 16 + 8. So, our new equation looks like this:
```
(2x + 4x) + (3y - 3y) = 16 + 8
```
Simplify the equation: Combine the like terms: 2x + 4x becomes 6x, and 3y - 3y becomes 0. On the right side, 16 + 8 equals 24. Our simplified equation is:
```
6x = 24
```
Solve for x: To isolate x, we divide both sides of the equation by 6:
```
6x / 6 = 24 / 6
x = 4
```
Great! We've found the value of x. It's 4. Now we need to find the value of y.
Substitute x into one of the original equations: We can use either of the original equations. Let's use the first one, 2x + 3y = 16. Substitute x = 4 into this equation:
```
2(4) + 3y = 16
```
Simplify and solve for y: Simplify the equation:
```
8 + 3y = 16
```
Subtract 8 from both sides:
```
3y = 16 - 8
3y = 8
```
Divide both sides by 3:
```
y = 8 / 3
```
So, the value of y is 8/3.
Write the solution as an ordered pair: The solution to the system of equations is x = 4 and y = 8/3. We write this as an ordered pair (x, y), so our solution is (4, 8/3).

And there you have it! We've successfully solved the system of equations using the elimination method. Let’s quickly recap our solution and then see how we can verify it to make sure we're on the right track.

Verifying the Solution

It's always a good idea to verify your solution to make sure you haven't made any mistakes along the way. To do this, we simply plug our values for x and y back into both of the original equations and see if they hold true. If both equations are satisfied, then we know we've found the correct solution. Let's give it a try:

Original Equations:
- 2x + 3y = 16
- 4x - 3y = 8
Our Solution:
- x = 4
- y = 8/3
Substitute into the first equation: Replace x with 4 and y with 8/3 in the first equation:
```
2(4) + 3(8/3) = 16
```
Simplify:
```
8 + 8 = 16
16 = 16
```
The first equation checks out!
Substitute into the second equation: Now let's do the same for the second equation:
```
4(4) - 3(8/3) = 8
```
Simplify:
```
16 - 8 = 8
8 = 8
```
The second equation also checks out!

Since our values for x and y satisfy both original equations, we can confidently say that our solution (4, 8/3) is correct. This verification step is super important, especially in exams or when you're working on complex problems. It's a quick way to catch any arithmetic errors or mistakes in your calculations. Now that we've verified our solution, let's zoom out a bit and discuss why understanding systems of equations is so valuable in the real world.

Real-World Applications of Systems of Equations

Okay, so we've mastered solving this system of equations, but you might be wondering,