Solving the System of Equations X - Y 2000 and X Y 450

Algebra is a powerful tool in mathematics, used to solve a wide range of problems. This article focuses on a specific problem involving a system of linear equations. The problem involves finding the values of X and Y given the following conditions:

Problem Statement

We are given the following equations:

X - Y 2000 X Y 450

Solution

We will proceed step by step to solve the given system of equations.

Substitution Method

The substitution method is often used when one of the equations is already solved for one variable. We can substitute the expression for X from the second equation into the first equation.

Substitute X from the second equation into the first equation: Y 450 - Y 2000 Combine like terms to simplify the equation: 450 2000 - Y Solve for Y: 2Y 2000 - 450 1550 Y 1550 / 2 775

Now that we have the value of Y, we can substitute it back into the second equation to find X:

X Y 450 X 775 450 1225

Thus, the values of X and Y are:

X 1225 Y 775

Verification

To ensure the correctness of our solution, let's verify the values by substituting them back into the original equations:

Check the first equation: 1225 - 775 2000 - True Check the second equation: 1225 775 450 - True

Conclusion

We have successfully solved the system of linear equations using the substitution method. The values of X and Y that satisfy both equations are X 1225 and Y 775.

Solving systems of equations not only helps in understanding algebraic manipulations but also in developing critical thinking skills. Whether you are a student, a teacher, or a professional, mastering algebra can greatly enhance your problem-solving abilities.