Solving Systems Of Equations By Combining Equations: A Step-by-Step Guide
In the realm of mathematics, particularly in algebra, solving systems of equations is a fundamental skill. Systems of equations represent situations where multiple equations with multiple variables need to be solved simultaneously. One powerful technique for tackling these systems is the method of combining equations, also known as elimination. This method involves manipulating the equations in the system in such a way that when they are added together, one of the variables is eliminated, leaving us with a single equation in one variable that can be easily solved. This article provides a detailed guide on how to solve the system of equations $-2x + y = 4$ and $4x - 6y = -48$ by combining the equations, along with a comprehensive explanation of the underlying principles and various applications of this technique.
Understanding the Concept of Combining Equations
The core idea behind combining equations lies in the principle that if we have two equations that are both true, then any linear combination of these equations will also be true. A linear combination simply means multiplying each equation by a constant and then adding the resulting equations together. The clever part is choosing the constants in such a way that the coefficients of one of the variables become opposites. When we add the equations, these terms cancel out, eliminating that variable and simplifying the system. This process allows us to reduce a two-variable system into a single-variable equation, which is much easier to solve.
For instance, consider the given system of equations:
Our goal is to eliminate either x or y. Observe that the coefficients of x are -2 and 4. We can make these coefficients opposites by multiplying the first equation by 2. This gives us -4x, which is the opposite of 4x in the second equation. Once we add the modified first equation to the second equation, the x terms will cancel out, leaving us with an equation in just y. This resulting equation can then be solved for y, and the value of y can be substituted back into either of the original equations to solve for x. This step-by-step approach makes the method of combining equations a systematic and effective technique for solving systems of linear equations.
Step-by-Step Solution to the System of Equations
Let's apply the method of combining equations to solve the system:
Step 1: Choose a Variable to Eliminate
In this case, we choose to eliminate x. As mentioned earlier, the coefficients of x are -2 and 4. To make them opposites, we can multiply Equation 1 by 2.
Step 2: Multiply One or Both Equations by a Constant
Multiply Equation 1 by 2:
Now we have:
Step 3: Add the Equations Together
Add the Modified Equation 1 to Equation 2:
Combine like terms:
Step 4: Solve for the Remaining Variable
Divide both sides by -4:
rac{-4y}{-4} = rac{-40}{-4}
Step 5: Substitute the Value Back into One of the Original Equations
Substitute y = 10 into Equation 1:
Subtract 10 from both sides:
Divide both sides by -2:
Step 6: Verify the Solution
Substitute x = 3 and y = 10 into both original equations to check if the solution is correct.
For Equation 1:
For Equation 2:
Since the solution satisfies both equations, the solution to the system is x = 3 and y = 10.
Therefore, the solution to the system of equations is (3, 10).
Alternative Approaches and Considerations
While the above steps outline the general process of solving systems of equations by combining equations, there can be alternative approaches and considerations depending on the specific system. For instance, instead of eliminating x first, we could have chosen to eliminate y. To do this, we would need to multiply Equation 1 by 6 to get a coefficient of 6y, and then add it to Equation 2, where the coefficient of y is -6. This would also lead to the same solution, but the intermediate steps would be slightly different. The choice of which variable to eliminate first often depends on which requires the least amount of manipulation of the equations.
Another consideration is when the coefficients of the variables don't have a clear common multiple. In such cases, it might be necessary to multiply both equations by different constants to ensure that the coefficients of one variable become opposites. For example, if we had the system 2x + 3y = 5 and 3x - 2y = 1, we could multiply the first equation by 3 and the second equation by -2 to eliminate x. This would result in the system 6x + 9y = 15 and -6x + 4y = -2, which can then be added to eliminate x.
It's also important to be aware of special cases. Sometimes, when combining equations, both variables are eliminated, resulting in an equation like 0 = 0. This indicates that the two original equations are essentially the same line, and the system has infinitely many solutions. Conversely, if combining equations leads to a contradiction, such as 0 = 5, it means that the system has no solution, and the lines represented by the equations are parallel.
Applications of Solving Systems of Equations
Solving systems of equations is not just an abstract mathematical exercise; it has numerous practical applications in various fields. Here are some examples:
- Physics: Systems of equations are used to solve problems involving motion, forces, and circuits. For example, in kinematics, you might have equations representing the position and velocity of an object as a function of time. Solving these equations simultaneously can help determine the object's trajectory.
- Engineering: Engineers use systems of equations to design structures, analyze circuits, and optimize processes. For instance, in structural engineering, systems of equations can be used to calculate the forces in different members of a truss.
- Economics: Economic models often involve systems of equations that represent supply, demand, and market equilibrium. Solving these systems can help economists predict prices, quantities, and other economic variables.
- Chemistry: Chemical reactions can be represented using systems of equations to balance chemical equations. This ensures that the number of atoms of each element is the same on both sides of the equation.
- Computer Science: Systems of equations are used in computer graphics, optimization algorithms, and machine learning. For example, in computer graphics, systems of equations can be used to transform objects in 3D space.
- Everyday Life: Many real-world problems can be modeled as systems of equations. For example, if you are trying to decide between two different phone plans with different pricing structures, you can set up a system of equations to determine which plan is more cost-effective based on your usage.
Conclusion
The method of combining equations is a powerful technique for solving systems of linear equations. By strategically manipulating the equations to eliminate one variable, we can reduce the system to a single equation in one variable, which can be easily solved. This method is widely applicable and is a fundamental skill in mathematics and various related fields. Understanding the underlying principles and practicing the steps involved can help you confidently solve a wide range of problems involving systems of equations. The solution to the system $-2x + y = 4$ and $4x - 6y = -48$ using this method is (3, 10), which demonstrates the effectiveness and efficiency of this technique. Whether you are a student learning algebra or a professional working in a technical field, mastering the method of combining equations is an invaluable asset for problem-solving and decision-making. Understanding the versatility and broad applicability of this method empowers individuals to tackle complex problems in a structured and logical manner. By reinforcing the importance of algebraic manipulation and careful substitution, this guide encourages a deeper appreciation for mathematical problem-solving techniques and their real-world implications.
