give me an essay on Systems of Linear Equations Sample Work

Question

GPT 3.5 · Answer

Systems of linear equations are a fundamental concept in algebra and are used in a variety of applications in mathematics, science, and engineering. A system of linear equations consists of two or more linear equations that share the same variables. The goal is to find the values of the variables that satisfy all of the equations simultaneously.

One method for solving systems of linear equations is the substitution method. This method involves solving one equation for one variable and then substituting that value into the other equations to find the remaining variables. For example, consider the system of equations:

2x + y = 7
3x - 2y = 10

We can solve the first equation for y in terms of x:

y = 7 - 2x

Then, we substitute this expression for y into the second equation:

3x - 2(7 - 2x) = 10
3x - 14 + 4x = 10
7x - 14 = 10
7x = 24
x = 24/7

Once we have found the value of x, we can substitute it back into the first equation to find the value of y:

2(24/7) + y = 7
48/7 + y = 7
y = 7 - 48/7
y = 7/7 - 48/7
y = -41/7

Therefore, the solution to the system of equations is x = 24/7 and y = -41/7.

Another method for solving systems of linear equations is the elimination method. This method involves adding or subtracting the equations to eliminate one of the variables. For example, consider the system of equations:

x + y = 5
2x - y = 1

We can add the two equations to eliminate y:

x + y + 2x - y = 5 + 1
3x = 6
x = 2

Then we can substitute the value of x back into one of the original equations to find the value of y:

2 + y = 5
y = 3

Therefore, the solution to the system of equations is x = 2 and y = 3.

In conclusion, systems of linear equations are important in mathematics and have various methods for solving them. The substitution and elimination methods are just a few examples of how we can find the solutions to these systems. By understanding these methods, we can apply them to solve real-world problems and make connections between different mathematical concepts.