Solving Systems of Equations: 20/x+1+4/y-1=5 and 10/x+1-4/y-1=1
In this article, we will solve a system of two equations involving variables x and y. The equations are:
Equation 1: 20/x+1+4/y-1=5 Equation 2: 10/x+1-4/y-1=1
Step 1: Simplify the Equations
Let's start by simplifying both equations by combining like terms:
Equation 1: 20/x + 4/y = 5 - 1 => 20/x + 4/y = 4 Equation 2: 10/x - 4/y = 1 - 1 => 10/x - 4/y = 0
Step 2: Solve the System of Equations
We can solve this system of equations using the method of substitution or elimination. In this case, we will use the elimination method.
First, we will multiply Equation 1 by 2 to make the coefficients of 1/x the same:
Equation 1 (multiplied by 2): 40/x + 8/y = 8
Now, we will add Equation 2 to the modified Equation 1 to eliminate the 1/x term:
(40/x + 8/y = 8) + (10/x - 4/y = 0)
Combine like terms:
48/x + 4/y = 8 -4/y + 4/y = 0 48/x = 8
Now, we can solve for x:
x = 48/8 => x = 6
Step 3: Substitute x into One of the Original Equations
Now that we have found x, we can substitute it into one of the original equations to find y. We will use Equation 1:
20/x + 4/y = 4 20/6 + 4/y = 4 10/3 + 4/y = 4
Subtract 10/3 from both sides:
4/y = 4 - 10/3 4/y = (12 - 10)/3 4/y = 2/3
Multiply both sides by y:
4 = 2y/3
Multiply both sides by 3:
12 = 2y
Divide both sides by 2:
y = 12/2 y = 6
Step 4: Write the Solution
The solution to the system of equations is x = 6 and y = 6.
Conclusion
In this article, we solved a system of two equations involving variables x and y. We used the elimination method to solve for x and then substituted x into one of the original equations to find y. The final solution is x = 6 and y = 6.
