Solving the System of Equations using Cross Multiplication
In this article, we will learn how to solve a system of equations using the cross multiplication method. The system of equations we will be solving is:
Equation 1: 2/x - 1 + 3/y + 1 = 2 Equation 2: 3/x - 1 + 2/y + 1 = 13/6
Step 1: Write the Equations in Standard Form
First, we need to write both equations in standard form, which means we need to get rid of the fractions. To do this, we can multiply both sides of each equation by the least common multiple (LCM) of the denominators.
Equation 1: 2x - x + 3y + y = 2xy (multiply both sides by xy) Equation 2: 3x - x + 2y + y = 13xy/6 (multiply both sides by 6xy)
Simplifying the equations, we get:
Equation 1: x + 3y = 2xy Equation 2: 3x + 2y = 13xy/6
Step 2: Use Cross Multiplication
Now, we can use cross multiplication to eliminate one of the variables. Let's eliminate y. We can do this by multiplying Equation 1 by 2 and Equation 2 by 3, and then equating the two expressions.
Equation 1: 2(x + 3y) = 2(2xy) Equation 2: 3(3x + 2y) = 3(13xy/6)
Expanding the equations, we get:
Equation 1: 2x + 6y = 4xy Equation 2: 9x + 6y = 13xy/2
Now, equating the two expressions, we get:
2x + 6y = 9x + 6y
Subtracting 2x and 6y from both sides, we get:
0 = 7x - 13xy/2
Step 3: Solve for x
Now, we can solve for x. Multiplying both sides by 2, we get:
0 = 14x - 13xy
Dividing both sides by -13, we get:
x = xy
Dividing both sides by y, we get:
x/y = 1
x = y
Step 4: Solve for y
Now that we have found x, we can find y by substituting x into one of the original equations. Let's use Equation 1.
2x - x + 3y + y = 2xy
Substituting x = y, we get:
2y - y + 3y + y = 2y^2
Simplifying the equation, we get:
6y = 2y^2
Dividing both sides by 2, we get:
3y = y^2
y^2 - 3y = 0
Factoring the equation, we get:
y(y - 3) = 0
This gives us two possible values for y: y = 0 or y = 3.
Substituting y = 3 into x = y, we get x = 3.
So, the solution to the system of equations is x = 3 and y = 3.
Conclusion
In this article, we learned how to solve a system of equations using the cross multiplication method. We applied this method to the system of equations 2/x - 1 + 3/y + 1 = 2 and 3/x - 1 + 2/y + 1 = 13/6, and found the solution to be x = 3 and y = 3.
