Solving Systems Of Linear Equations Finding The Solution (a C)

Jul 31, 2025 by ADMIN 63 views

Hey everyone! Today, let's dive into the fascinating world of linear equations and explore how to find solutions for systems of equations. Specifically, we're going to tackle a problem where we need to determine the solution (a, c) for a given system. So, grab your thinking caps, and let's get started!

The System of Equations

We're presented with the following system of linear equations:

2a - 3c = -6
a + 2c = 11

Our mission, should we choose to accept it (and we do!), is to find the values of 'a' and 'c' that satisfy both of these equations simultaneously. There are a few ways we can approach this, but we'll focus on two popular methods: substitution and elimination.

Method 1: Substitution - A Step-by-Step Approach

Substitution involves solving one equation for one variable and then substituting that expression into the other equation. This effectively reduces the system to a single equation with a single variable, which we can then solve. Let's break it down step by step:

Choose an equation and solve for one variable: Looking at our system, the second equation (a + 2c = 11) seems easier to manipulate. Let's solve it for 'a':
```
a = 11 - 2c
```
Substitute the expression into the other equation: Now that we have an expression for 'a', we can substitute it into the first equation (2a - 3c = -6):
```
2(11 - 2c) - 3c = -6
```
Simplify and solve for the remaining variable: Let's simplify the equation and solve for 'c':
```
22 - 4c - 3c = -6
22 - 7c = -6
-7c = -28
c = 4
```
Great! We've found that c = 4.
Substitute the value back to find the other variable: Now that we know 'c', we can substitute it back into either of the original equations or the expression we found for 'a' (a = 11 - 2c). Let's use the latter:
```
a = 11 - 2(4)
a = 11 - 8
a = 3
```
So, we've found that a = 3.

Method 2: Elimination - The Power of Cancellation

Elimination involves manipulating the equations so that when you add or subtract them, one of the variables cancels out. This leaves you with a single equation with a single variable, just like in substitution. Here's how it works for our system:

Multiply equations to match coefficients: We want to make the coefficients of either 'a' or 'c' the same (but with opposite signs) in both equations. Let's target the 'a' variable. Multiply the second equation (a + 2c = 11) by -2:
```
-2(a + 2c) = -2(11)
-2a - 4c = -22
```
Now our system looks like this:
```
2a - 3c = -6
-2a - 4c = -22
```
Add the equations: Now, add the two equations together. Notice how the '2a' and '-2a' terms cancel out:
```
(2a - 3c) + (-2a - 4c) = -6 + (-22)
-7c = -28
```
Solve for the remaining variable: We're left with an equation in terms of 'c'. Solve for 'c':
```
c = 4
```
Again, we find that c = 4.
Substitute back to find the other variable: Just like in substitution, we substitute the value of 'c' back into one of the original equations to find 'a'. Let's use the second equation (a + 2c = 11):
```
a + 2(4) = 11
a + 8 = 11
a = 3
```
And we confirm that a = 3.

The Solution

Using both the substitution and elimination methods, we've arrived at the same solution: a = 3 and c = 4. Therefore, the solution to the system of linear equations is the ordered pair (3, 4).

Checking Our Work - A Crucial Step

It's always a good idea to check our solution to make sure we haven't made any mistakes. We can do this by substituting the values of 'a' and 'c' back into the original equations:

Equation 1: 2a - 3c = -6

2(3) - 3(4) = 6 - 12 = -6  (Checks out!)

Equation 2: a + 2c = 11
```
3 + 2(4) = 3 + 8 = 11  (Checks out!)
```

Since our solution satisfies both equations, we can be confident that (3, 4) is indeed the correct solution.

Why This Matters - Applications of Linear Systems

You might be wondering, "Okay, we solved for 'a' and 'c', but what's the big deal?" Well, systems of linear equations pop up in all sorts of real-world scenarios. Here are just a few examples:

Mixing problems: Imagine you're blending two different solutions with varying concentrations to get a desired final concentration. Systems of equations can help you determine how much of each solution to use.
Distance, rate, and time problems: If you have two objects moving at different speeds and want to know when and where they'll meet, you can set up a system of equations.
Supply and demand: In economics, the equilibrium point where supply and demand curves intersect can be found by solving a system of equations.
Circuit analysis: Electrical engineers use systems of equations to analyze the flow of current in circuits.
Computer graphics: Linear algebra, which includes systems of equations, is fundamental to creating 3D graphics and animations.

The ability to solve systems of linear equations is a valuable skill in many fields, making it a fundamental concept in mathematics.

Common Mistakes to Avoid

Solving systems of equations can be tricky, and it's easy to make small errors that lead to incorrect solutions. Here are some common mistakes to watch out for:

Sign errors: Pay close attention to signs when multiplying or distributing, especially when dealing with negative numbers.
Arithmetic errors: Double-check your calculations, especially when adding or subtracting terms.
Incorrect substitution: Make sure you're substituting the expression or value into the correct equation.
Forgetting to solve for both variables: Remember, the solution is an ordered pair (or a set of values if you have more variables), so you need to find the value of each variable.
Not checking your solution: Always take the time to check your solution by substituting it back into the original equations. This can help you catch errors and ensure your answer is correct.

By being mindful of these common mistakes, you can improve your accuracy and confidence in solving systems of equations.

Conclusion: Mastering the Art of Solving

So, there you have it! We've successfully navigated the world of linear equations and found the solution (a, c) = (3, 4) for our given system. We explored two powerful methods – substitution and elimination – and saw how they can be used to solve these types of problems. Remember, practice makes perfect, so keep honing your skills, and you'll become a master equation solver in no time!

Understanding how to solve systems of linear equations is not just about getting the right answer; it's about developing a logical and systematic approach to problem-solving. These skills will serve you well in many areas of mathematics and beyond.

Keep practicing, keep exploring, and most importantly, keep having fun with math! You guys got this!