Solving a System of Equations: 57/x + y + 6/x - y = 5 and 38/x + y + 21/x - y = 9
In this article, we will solve a system of equations involving two variables, x and y. The system consists of two equations:
Equation 1: 57/x + y + 6/x - y = 5 Equation 2: 38/x + y + 21/x - y = 9
Our goal is to find the values of x and y that satisfy both equations.
Simplifying the Equations
Let's simplify both equations by combining like terms:
Equation 1: (57 + 6)/x + y - y = 5 Equation 1: 63/x = 5
Equation 2: (38 + 21)/x + y - y = 9 Equation 2: 59/x = 9
Solving the System
Now we have a simpler system of equations:
Equation 1: 63/x = 5 Equation 2: 59/x = 9
We can solve for x by multiplying both equations by x:
Equation 1: 63 = 5x Equation 2: 59 = 9x
Now, divide both equations by the coefficient of x:
Equation 1: x = 63/5 Equation 2: x = 59/9
Since both equations must be true, we can set them equal to each other:
63/5 = 59/9
Solving for x
To solve for x, we can cross-multiply:
63 × 9 = 59 × 5 567 = 295
Now, divide both sides by 295:
x = 567/295 x = 63/35 x = 9/5
Solving for y
Now that we have the value of x, we can substitute it into one of the original equations to solve for y. Let's use Equation 1:
57/x + y + 6/x - y = 5 57/(9/5) + y + 6/(9/5) - y = 5
Simplify the equation:
57/(9/5) = 35 6/(9/5) = 10/3
Now, substitute the values:
35 + y + 10/3 - y = 5
Simplify further:
35 + 10/3 = 5 35 = 5 - 10/3 35 = 5 - 10/3 35 = 15/3 35 × 3 = 15 105 = 15
Now, subtract 105 from both sides:
y = -90
Final Solution
The final solution to the system of equations is:
x = 9/5 y = -90
Therefore, the values of x and y that satisfy both equations are x = 9/5 and y = -90.
