Solving Simultaneous Equations: xy 2 and x - y 5

In this article, we will explore how to solve the system of equations given by:

These types of equations, where both the product and the difference of two variables are given, are known as simultaneous equations. There are multiple methods to solve such systems, including substitution and elimination.

Method of Elimination

To solve the given equations using the method of elimination, we can begin by examining the equations:

xy  2  Equation 1x - y  5  Equation 2

The first step is to add both equations to eliminate the variable y:

xy   x - y  2   52x  7

Now, solve for x:

x  frac{7}{2}  3.5

With the value of x known, we can substitute it back into one of the original equations to find y. We'll use Equation 1 for convenience:

3.5   y  2y  2 - 3.5  -1.5

Thus, the solution to the system of equations is:

x  3.5y  -1.5

Algebraic Manipulation

We can also approach the problem algebraically by manipulating the given equations. Starting with:

xy 2 …..(1)

x - y 5 …..(2)

From Equation 2, we can express x in terms of y:

x  5   y

Substitute this expression into Equation 1:

(5   y)y  25y   y^2  2

Now we have a quadratic equation in terms of y:

y^2   5y - 2  0

To solve this quadratic equation, we can use the quadratic formula:

y  frac{-b pm sqrt{b^2 - 4ac}}{2a}

Here, a 1, b 5, and c -2:

y  frac{-5 pm sqrt{5^2 - 4(1)(-2)}}{2(1)}y  frac{-5 pm sqrt{25   8}}{2}y  frac{-5 pm sqrt{33}}{2}

This results in two potential values for y:

y_1  frac{-5   sqrt{33}}{2} approx -1.5y_2  frac{-5 - sqrt{33}}{2} approx -3.5

Substituting these back into x 5 y, we find:

x_1  5   (-1.5)  3.5x_2  5   (-3.5)  1.5

Thus, the solutions are:

x  3.5, y  -1.5x  1.5, y  -3.5

These solutions demonstrate how to solve the system of equations using both graphical and algebraic methods.

Conclusion

The key to solving these types of equations is to identify the appropriate method and apply it consistently. Understanding the principles of algebraic manipulation and the method of elimination is crucial for solving such problems effectively. Whether you use the substitution or elimination method, the goal is to find the unique values of x and y that satisfy both equations.

If you are looking to improve your skills in solving simultaneous equations, practice is key. Try solving similar problems and explore different methods to enhance your problem-solving abilities in algebra.