Solving Linear Equations Finding The Solution Q R

Jul 9, 2025 by ADMIN 50 views

Solving Linear Equations A Comprehensive Guide to Finding the Solution (q, r)

In the realm of mathematics, solving systems of linear equations is a fundamental skill. These systems arise in diverse applications, from modeling real-world phenomena to optimizing complex processes. This article delves into the step-by-step process of solving a given system of linear equations, providing a comprehensive guide for readers to grasp the underlying concepts and master the techniques involved. We will use the following system of equations as an example:

12q + 3r = 15
-4q - 4r = -44

By the end of this guide, you'll be well-equipped to tackle similar problems and confidently determine the solutions to linear equation systems. Linear equations, at their core, represent relationships between variables where the highest power of any variable is one. A system of linear equations is simply a collection of two or more such equations that share the same variables. The solution to a system of linear equations is a set of values for the variables that simultaneously satisfy all equations in the system. In simpler terms, it's the point where all the lines represented by the equations intersect.

Understanding Linear Equations and Systems

Before diving into the solution, it's crucial to grasp the fundamentals of linear equations. A linear equation is an algebraic expression in which the highest power of any variable is 1. For instance, 12q + 3r = 15 and -4q - 4r = -44 are both linear equations. A system of linear equations is a set of two or more linear equations that share the same variables. Solving a system of linear equations means finding the values for the variables that satisfy all equations simultaneously. Geometrically, each linear equation in two variables represents a straight line. The solution to a system of two linear equations in two variables is the point where the two lines intersect. If the lines are parallel, there is no solution. If the lines coincide, there are infinitely many solutions. Understanding the graphical representation helps visualize the concept of solutions. The core concept revolves around finding values for the variables that make all equations in the system true at the same time. Each equation represents a constraint, and the solution is the point where all constraints are satisfied. Think of it as finding the common ground where all the equations agree.

Methods for Solving Systems of Linear Equations

There are several methods to solve systems of linear equations, each with its strengths and weaknesses. The most common methods include:

Substitution: This method involves solving one equation for one variable and substituting that expression into the other equation. This eliminates one variable, leaving a single equation in one variable that can be easily solved. The value obtained is then substituted back into one of the original equations to find the value of the other variable. The substitution method is particularly effective when one of the equations can be easily solved for one variable in terms of the other. For example, if one equation is y = 2x + 1, substituting this expression for y into the other equation simplifies the problem.
Elimination (or Addition): This method involves manipulating the equations so that the coefficients of one variable are opposites. Adding the equations then eliminates that variable, leaving a single equation in one variable. This method is advantageous when the coefficients of one variable are already opposites or can be easily made opposites by multiplying one or both equations by a constant. The elimination method is especially useful when dealing with equations in standard form (Ax + By = C).
Graphing: This method involves graphing each equation on the same coordinate plane. The point of intersection of the lines represents the solution to the system. While visually intuitive, graphing may not always provide precise solutions, especially if the intersection point has non-integer coordinates. Graphing is most suitable for systems with simple equations and when an approximate solution is sufficient.
Matrix Methods: These methods, including Gaussian elimination and matrix inversion, are more advanced techniques suitable for larger systems of equations. Matrix methods provide a systematic and efficient way to solve complex systems, particularly those with many variables. These methods are often used in computer algorithms for solving linear systems.

Step-by-Step Solution Using the Elimination Method

For our example system,

12q + 3r = 15
-4q - 4r = -44

let's use the elimination method. This method is particularly well-suited for this system because we can easily eliminate the q variable by multiplying the second equation by 3. The elimination method focuses on strategically manipulating the equations to eliminate one variable, simplifying the system. It's a powerful technique that relies on the principle that adding or subtracting equal quantities from both sides of an equation maintains the equality.

Step 1: Multiply the second equation by 3:

This gives us:

-12q - 12r = -132

The goal here is to make the coefficients of either q or r opposites in the two equations. By multiplying the second equation by 3, we've achieved this for the q variable. Now, the coefficients of q are 12 and -12, which are opposites.

Step 2: Add the modified second equation to the first equation:

Adding the equations:

12q + 3r = 15
-12q - 12r = -132

Results in:

-9r = -117

The q terms cancel out, leaving us with a single equation in terms of r. This is the key step in the elimination method – reducing the system to a single equation with one unknown. This resulting equation is much easier to solve.

Step 3: Solve for r:

Divide both sides by -9:

r = 13

We've now found the value of r. This value satisfies the combined equation and is a crucial part of the solution to the system. This step demonstrates the power of algebraic manipulation in isolating variables and finding their values.

Step 4: Substitute the value of r back into one of the original equations to solve for q:

Let's use the first equation:

12q + 3(13) = 15

Substituting the value of r back into one of the original equations allows us to solve for the remaining variable, q. This step is essential to find the complete solution to the system. Choosing the simpler equation can often make the calculation easier.

Step 5: Simplify and solve for q:

12q + 39 = 15
12q = -24
q = -2

We've now found the value of q. This completes the solution process. The values of q and r we've found simultaneously satisfy both equations in the original system.

The Solution

Therefore, the solution to the system of equations is (q, r) = (-2, 13). This corresponds to option B in the given choices. Verifying the solution is a crucial step to ensure accuracy. To verify, substitute the values of q and r back into both original equations and check if they hold true. This step helps catch any potential errors made during the solution process.

Verification

Let's verify our solution by substituting q = -2 and r = 13 into the original equations:

Equation 1:

12(-2) + 3(13) = -24 + 39 = 15

This equation holds true.

Equation 2:

-4(-2) - 4(13) = 8 - 52 = -44

This equation also holds true. Since the solution satisfies both equations, we have confirmed its correctness. This verification step is a cornerstone of problem-solving in mathematics, ensuring that the obtained solution is accurate and reliable.

Conclusion

Solving systems of linear equations is a fundamental skill in mathematics with applications across various fields. This guide has demonstrated a step-by-step approach using the elimination method to find the solution to a specific system of equations. By understanding the underlying concepts and mastering the techniques, readers can confidently tackle similar problems and gain a deeper appreciation for the power of linear algebra. The key takeaways from this guide are the importance of understanding the different methods for solving linear systems, the step-by-step application of the elimination method, and the crucial role of verification in ensuring the accuracy of the solution. Practice is essential for mastering these techniques and developing problem-solving skills in mathematics. By working through various examples and applying the methods discussed, you can build confidence and proficiency in solving linear equation systems.