Solving System Of Equations -5x + 7y = 31 And 3x + 4y = 6
In mathematics, solving systems of equations is a fundamental skill with applications spanning various fields, from engineering and physics to economics and computer science. A system of equations is a set of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that satisfy all equations simultaneously. One of the most effective methods for solving systems of linear equations is the elimination method, also known as the addition method. This method involves manipulating the equations to eliminate one variable, allowing us to solve for the remaining variable. In this article, we will delve into the elimination method, providing a step-by-step guide with examples and addressing potential scenarios such as inconsistent and dependent systems.
Understanding the Elimination Method
The core principle behind the elimination method is to strategically manipulate the equations in the system so that when they are added together, one of the variables cancels out. This is achieved by multiplying one or both equations by a constant factor so that the coefficients of one of the variables are opposites (e.g., 3x and -3x). Once we eliminate a variable, we are left with a single equation in one variable, which can be easily solved. The solution for that variable is then substituted back into one of the original equations to solve for the other variable. Let's illustrate this with an example:
Step-by-Step Guide to the Elimination Method
Consider the following system of equations:
-5x + 7y = 31
3x + 4y = 6
Our goal is to find the values of x and y that satisfy both equations.
Step 1: Choose a Variable to Eliminate
First, we need to decide which variable to eliminate. In this case, let's choose to eliminate x. To do this, we need to make the coefficients of x in both equations opposites.
Step 2: Multiply Equations by Suitable Constants
The coefficient of x in the first equation is -5, and in the second equation, it is 3. To make these opposites, we can multiply the first equation by 3 and the second equation by 5:
3 * (-5x + 7y) = 3 * 31
5 * (3x + 4y) = 5 * 6
This gives us:
-15x + 21y = 93
15x + 20y = 30
Notice that the coefficients of x are now -15 and 15, which are opposites.
Step 3: Add the Equations
Now, we add the two equations together:
(-15x + 21y) + (15x + 20y) = 93 + 30
The -15x and 15x terms cancel out, leaving us with:
41y = 123
Step 4: Solve for the Remaining Variable
Divide both sides by 41 to solve for y:
y = 123 / 41
y = 3
Step 5: Substitute the Value Back into One of the Original Equations
Now that we have the value of y, we can substitute it back into either of the original equations to solve for x. Let's use the second equation:
3x + 4y = 6
Substitute y = 3:
3x + 4(3) = 6
3x + 12 = 6
Step 6: Solve for the Other Variable
Subtract 12 from both sides:
3x = -6
Divide by 3:
x = -2
Step 7: Write the Solution
The solution to the system of equations is x = -2 and y = 3. We can write this as an ordered pair: (-2, 3).
Verification of the Solution
It's always a good practice to verify the solution by substituting the values of x and y back into both original equations to ensure they are satisfied:
For the first equation:
-5x + 7y = 31
-5(-2) + 7(3) = 10 + 21 = 31 (Correct)
For the second equation:
3x + 4y = 6
3(-2) + 4(3) = -6 + 12 = 6 (Correct)
Since the solution satisfies both equations, we have successfully solved the system.
Inconsistent and Dependent Systems
Not all systems of equations have a unique solution. Some systems may have no solution (inconsistent), while others may have infinitely many solutions (dependent). Let's explore these scenarios:
Inconsistent Systems
An inconsistent system of equations is a system that has no solution. This occurs when the equations represent parallel lines that never intersect. When using the elimination method, an inconsistent system will result in a contradiction, such as 0 = a non-zero number.
Consider the following system:
x + y = 3
x + y = 5
If we try to solve this system using the elimination method, we can multiply the first equation by -1:
-x - y = -3
Now, add the modified first equation to the second equation:
(-x - y) + (x + y) = -3 + 5
0 = 2
This is a contradiction. The equation 0 = 2 is never true, indicating that the system is inconsistent and has no solution. In graphical terms, these equations represent parallel lines.
Dependent Systems
A dependent system of equations is a system that has infinitely many solutions. This occurs when the equations represent the same line or multiples of each other. When using the elimination method, a dependent system will result in an identity, such as 0 = 0.
Consider the following system:
2x + y = 4
4x + 2y = 8
Notice that the second equation is simply the first equation multiplied by 2. This means the equations represent the same line. Let's use the elimination method to see what happens. Multiply the first equation by -2:
-4x - 2y = -8
Add the modified first equation to the second equation:
(-4x - 2y) + (4x + 2y) = -8 + 8
0 = 0
This is an identity. The equation 0 = 0 is always true, indicating that the system is dependent and has infinitely many solutions. Any point on the line represented by either equation is a solution to the system. To express the general solution, we can solve one of the equations for one variable in terms of the other. For example, from the first equation, we can write:
y = 4 - 2x
So, the general solution can be written as (x, 4 - 2x), where x can be any real number.
Applications of Solving Systems of Equations
The ability to solve systems of equations is crucial in many real-world applications. Here are a few examples:
1. Engineering:
Engineers often use systems of equations to analyze circuits, structural designs, and fluid dynamics. For instance, in circuit analysis, Kirchhoff's laws lead to systems of equations that can be solved to determine the currents and voltages in different parts of the circuit. Structural engineers use systems of equations to calculate the forces and stresses in a structure, ensuring its stability and safety. Similarly, in fluid dynamics, systems of equations describe the flow of fluids, which is essential for designing pipelines, aircraft, and hydraulic systems.
2. Economics:
In economics, systems of equations are used to model supply and demand, market equilibrium, and macroeconomic relationships. For example, the equilibrium price and quantity in a market can be found by solving a system of equations representing the supply and demand curves. Macroeconomic models often involve systems of equations that describe the relationships between variables such as GDP, inflation, unemployment, and interest rates. These models help economists analyze economic trends and formulate policy recommendations.
3. Physics:
Physics relies heavily on systems of equations to describe the motion of objects, the behavior of waves, and the interactions of particles. In classical mechanics, Newton's laws of motion lead to systems of equations that can be solved to predict the trajectory of a projectile or the motion of a pendulum. In electromagnetism, Maxwell's equations form a system of partial differential equations that describe the behavior of electric and magnetic fields. Solving these equations is essential for designing antennas, waveguides, and other electromagnetic devices. Quantum mechanics also involves solving systems of equations, such as the Schrödinger equation, to determine the behavior of atoms and subatomic particles.
4. Computer Science:
Systems of equations are used in computer graphics, optimization algorithms, and network analysis. In computer graphics, systems of equations are used to perform transformations such as scaling, rotation, and translation of objects. Optimization algorithms, such as linear programming, involve solving systems of equations to find the optimal solution to a problem. Network analysis uses systems of equations to model the flow of data in a network, which is essential for designing efficient communication systems.
5. Chemistry:
In chemistry, systems of equations are used in stoichiometry, chemical kinetics, and equilibrium calculations. Stoichiometry involves calculating the amounts of reactants and products in chemical reactions, which often requires solving systems of equations. Chemical kinetics deals with the rates of chemical reactions, and the rate equations can form a system of differential equations that need to be solved. Equilibrium calculations involve determining the concentrations of reactants and products at equilibrium, which can be done by solving a system of equations derived from the equilibrium constant.
Conclusion
The elimination method is a powerful tool for solving systems of linear equations. By strategically manipulating the equations to eliminate one variable, we can find the solution efficiently. However, it's important to recognize inconsistent and dependent systems, which indicate either no solution or infinitely many solutions, respectively. The ability to solve systems of equations is a fundamental skill with wide-ranging applications in various fields, making it an essential topic in mathematics education and beyond. Mastering this technique provides a strong foundation for tackling more complex mathematical problems and real-world challenges. Remember to always verify your solutions to ensure accuracy and a thorough understanding of the concepts involved.
