Solving Linear Equations Finding The Correct Ordered Pair

Hey guys! Ever found yourself staring at a system of equations, feeling like you're trying to solve a puzzle with missing pieces? Well, you're not alone! Linear equations might seem intimidating at first, but with a few tricks up your sleeve, you can crack them like a pro. Let's dive into one such puzzle today and find the ordered pair that fits just right.

Understanding the System of Linear Equations

When dealing with linear equations, it's essential to grasp the fundamentals. A linear equation, in its simplest form, represents a straight line on a graph. When you have a system of linear equations, you're essentially looking at two or more lines and trying to find the point where they intersect. This intersection point, represented as an ordered pair (a, b), is the solution that satisfies all the equations in the system.

In our case, we have the following system:

-2a + 3b = 14
a - 4b = 3

Our mission, should we choose to accept it (and we do!), is to find the values of 'a' and 'b' that make both of these equations true simultaneously. Think of it like finding the perfect blend of ingredients in a recipe – too much of one, and the whole dish is off!

Methods to Solve Systems of Linear Equations

There are several ways to tackle these systems, but two common methods are substitution and elimination. Let's briefly touch on these before we dive into solving our specific problem.

• Substitution: This method involves solving one equation for one variable and then substituting that expression into the other equation. This leaves you with a single equation with one variable, which is much easier to solve. Once you find the value of one variable, you can plug it back into either of the original equations to find the value of the other.
• Elimination: The elimination method focuses on canceling out one of the variables by adding or subtracting the equations. This often involves multiplying one or both equations by a constant to make the coefficients of one variable match (or be opposites). When you add or subtract the equations, that variable disappears, leaving you with a single equation in one variable.

Cracking the Code: Solving the System

Now, let's get our hands dirty and solve the system we have. For this particular problem, we'll use the substitution method, as it seems like a straightforward approach.

Looking at the second equation, a - 4b = 3, it seems easiest to solve for 'a'. We can simply add 4b to both sides to get:

a = 4b + 3

Now we have an expression for 'a' in terms of 'b'. The next step is to substitute this expression into the first equation:

-2a + 3b = 14
-2(4b + 3) + 3b = 14

See what we did there? We replaced 'a' with (4b + 3). Now we have an equation with only one variable, 'b'. Let's simplify and solve for 'b':

-8b - 6 + 3b = 14
-5b - 6 = 14
-5b = 20
b = -4

Alright! We've found the value of 'b'! Now that we know b = -4, we can plug it back into our expression for 'a':

a = 4b + 3
a = 4(-4) + 3
a = -16 + 3
a = -13

Boom! We've got both 'a' and 'b'! So, our solution is the ordered pair (-13, -4).

Verifying the Solution

But wait, we're not quite done yet. It's always a good idea to verify our solution to make sure we didn't make any sneaky errors along the way. To do this, we'll plug our values of 'a' and 'b' into both original equations and see if they hold true.

Let's start with the first equation:

-2a + 3b = 14
-2(-13) + 3(-4) = 14
26 - 12 = 14
14 = 14

Check! The first equation is satisfied. Now let's try the second equation:

a - 4b = 3
-13 - 4(-4) = 3
-13 + 16 = 3
3 = 3

Double-check! The second equation is also satisfied. This confirms that our solution, (-13, -4), is indeed correct.

The Answer and Why It Matters

So, the ordered pair (-13, -4) is the solution to the given system of linear equations. This corresponds to option A in the choices provided.

But why does this matter? Well, systems of linear equations pop up in all sorts of real-world scenarios. Think about mixing solutions in a chemistry lab, planning a budget, or even optimizing traffic flow. Understanding how to solve these systems is a valuable skill that can help you make informed decisions and solve problems in various fields.

Exploring Other Solution Methods

While we used the substitution method in this case, remember that the elimination method is another powerful tool in your arsenal. Let's briefly explore how we could have used elimination to solve this same system.

Our system again is:

-2a + 3b = 14
a - 4b = 3

To use elimination, we want to make the coefficients of either 'a' or 'b' opposites. Let's focus on 'a'. We can multiply the second equation by 2 to get:

2(a - 4b) = 2(3)
2a - 8b = 6

Now our system looks like this:

-2a + 3b = 14
2a - 8b = 6

Notice that the coefficients of 'a' are now opposites (-2 and 2). If we add these two equations together, the 'a' terms will cancel out:

(-2a + 3b) + (2a - 8b) = 14 + 6
-5b = 20
b = -4

Just like before, we find that b = -4. Now we can plug this value back into either of the original equations (or the modified one) to solve for 'a'. Let's use the second original equation:

a - 4b = 3
a - 4(-4) = 3
a + 16 = 3
a = -13

And voila! We arrive at the same solution, (-13, -4), using a different method. This highlights the beauty of mathematics – there's often more than one way to reach the same destination!

Tips and Tricks for Solving Linear Equations

Before we wrap up, here are a few extra tips and tricks to keep in mind when tackling systems of linear equations:

• Stay Organized: Keep your work neat and tidy. Write down each step clearly, and double-check your calculations. A small mistake can throw off the entire solution.
• Choose the Right Method: Sometimes, one method (substitution or elimination) is clearly easier than the other. Look at the equations and see which method will lead to fewer steps and simpler calculations.
• Don't Be Afraid to Multiply: Multiplying equations by a constant is a powerful tool for both substitution and elimination. It allows you to manipulate the equations to make the variables line up nicely.
• Check Your Work: Always, always, always check your solution by plugging it back into the original equations. This will catch any errors and give you confidence in your answer.
• Practice Makes Perfect: Like any skill, solving linear equations gets easier with practice. The more problems you solve, the more comfortable you'll become with the different techniques and strategies.

Conclusion: You've Cracked the Code!

So, there you have it! We've successfully navigated the world of linear equations and found the ordered pair that solves our system. Remember, guys, math might seem like a daunting maze, but with a bit of logic, a dash of strategy, and a whole lot of practice, you can conquer any equation that comes your way. Keep exploring, keep learning, and most importantly, keep having fun with math! You've got this!