Solving systems of linear equations is a fundamental concept in mathematics, with applications spanning various fields, from engineering and physics to economics and computer science. In this comprehensive guide, we will delve into the methods for solving systems of linear equations and apply these techniques to determine the specific solution (a, c) for the given system:
2a - 3c = -6
a + 2c = 11
We will explore the concepts of substitution and elimination, providing step-by-step explanations and examples to ensure a clear understanding. By the end of this article, you will be equipped with the knowledge and skills necessary to confidently solve similar systems of linear equations.
Understanding Systems of Linear Equations
At its core, a system of linear equations is a collection of two or more linear equations that share the same set of variables. A linear equation, in its simplest form, represents a straight line when graphed on a coordinate plane. The solution to a system of linear equations is the point (or set of points) that satisfies all equations simultaneously. Geometrically, this corresponds to the point(s) where the lines intersect.
In our case, we have two linear equations with two variables, 'a' and 'c'. The goal is to find the values of 'a' and 'c' that make both equations true. This pair of values, denoted as (a, c), represents the solution to the system.
Methods for Solving Linear Equations
There are several methods for solving systems of linear equations, each with its own advantages and suitability for different types of systems. The two most commonly used methods are:
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Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation. This eliminates one variable, resulting in a single equation with one unknown, which can then be easily solved. The value obtained is then substituted back into one of the original equations to find the value of the other variable.
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Elimination Method: Also known as the addition or subtraction method, this technique involves manipulating the equations to make the coefficients of one variable the same (or additive inverses) in both equations. By adding or subtracting the equations, one variable is eliminated, leaving a single equation with one unknown. Solving this equation gives the value of one variable, which can then be substituted back into one of the original equations to find the value of the other variable.
Step-by-Step Solution Using the Substitution Method
Let's apply the substitution method to solve the given system of equations:
2a - 3c = -6 (Equation 1)
a + 2c = 11 (Equation 2)
Step 1: Solve one equation for one variable.
We can choose either equation and solve for either variable. For simplicity, let's solve Equation 2 for 'a':
a = 11 - 2c
Step 2: Substitute the expression into the other equation.
Now, substitute this expression for 'a' into Equation 1:
2(11 - 2c) - 3c = -6
Step 3: Solve for the remaining variable.
Simplify and solve for 'c':
22 - 4c - 3c = -6
22 - 7c = -6
-7c = -28
c = 4
Step 4: Substitute back to find the other variable.
Substitute the value of 'c' (4) back into the expression for 'a':
a = 11 - 2(4)
a = 11 - 8
a = 3
Therefore, the solution to the system of equations using the substitution method is (a, c) = (3, 4).
Step-by-Step Solution Using the Elimination Method
Now, let's solve the same system of equations using the elimination method:
2a - 3c = -6 (Equation 1)
a + 2c = 11 (Equation 2)
Step 1: Manipulate the equations to match coefficients.
To eliminate 'a', we can multiply Equation 2 by -2:
-2(a + 2c) = -2(11)
-2a - 4c = -22 (Equation 3)
Step 2: Add or subtract the equations to eliminate a variable.
Now, add Equation 1 and Equation 3:
(2a - 3c) + (-2a - 4c) = -6 + (-22)
-7c = -28
Step 3: Solve for the remaining variable.
Solve for 'c':
c = 4
Step 4: Substitute back to find the other variable.
Substitute the value of 'c' (4) back into either Equation 1 or Equation 2. Let's use Equation 2:
a + 2(4) = 11
a + 8 = 11
a = 3
Thus, the solution to the system of equations using the elimination method is also (a, c) = (3, 4).
Verifying the Solution
To ensure the accuracy of our solution, it's crucial to verify that the values (a, c) = (3, 4) satisfy both equations in the original system. Let's substitute these values into the equations:
For Equation 1:
2a - 3c = -6
2(3) - 3(4) = -6
6 - 12 = -6
-6 = -6 (True)
For Equation 2:
a + 2c = 11
3 + 2(4) = 11
3 + 8 = 11
11 = 11 (True)
Since the values (a, c) = (3, 4) satisfy both equations, we can confidently conclude that this is the correct solution to the system of linear equations.
Conclusion
In this comprehensive guide, we have explored the concept of solving systems of linear equations and demonstrated the application of two primary methods: substitution and elimination. By applying these techniques, we successfully determined the solution (a, c) = (3, 4) for the given system of equations. Understanding and mastering these methods is essential for various mathematical and scientific applications.
Remember, the key to solving systems of linear equations lies in carefully applying the chosen method and verifying the solution to ensure accuracy. With practice and a solid understanding of the underlying principles, you can confidently tackle any system of linear equations that comes your way.
Key Takeaways:
- Systems of linear equations represent a fundamental concept in mathematics.
- The solution to a system of linear equations is the point that satisfies all equations simultaneously.
- The substitution and elimination methods are two primary techniques for solving these systems.
- Verification of the solution is crucial to ensure accuracy.
- Mastering these methods opens doors to various applications in diverse fields.
By understanding the principles and practicing the techniques outlined in this article, you can confidently solve systems of linear equations and apply this knowledge to various real-world problems. Keep exploring, keep learning, and keep mastering the power of mathematics!
