Cross is a method of solving equations with fractions. It’s a simple and quick technique that enables us to solve equations without having to rewrite them in the same denominator. Cross multiplication is based on the property that the product of the numerator of one fraction and the denominator of the other is equal to the product of the numerator of the other fraction and the denominator of the first.

To understand cross multiplication, let us first take an example. Suppose we have the following equation:

3/4 = x/8

To solve this, we can use cross multiplication method as follows:

3 * 8 = 4 * x
24 = 4x

Now we can easily solve for x by dividing both sides by 4:

x = 6

So x = 6 is the solution to the equation 3/4 = x/8.

Let’s look at another example:

2/3 = 4/y

We can solve for y using cross multiplication as follows:

2 * y = 3 * 4
2y = 12

Now we can solve for y by dividing both sides by 2:

y = 6

So y = 6 is the solution to the equation 2/3 = 4/y.

Now we will look at a more complex example:

3/x + 2/y = 5/6

To solve this equation using cross multiplication, we first multiply both sides by the least common denominator of the fractions, which is 6xy:

6y * 3/x + 6x * 2/y = 6xy * 5/6

Simplifying this expression, we get:

18y/x + 12x/y = 5xy

Now we can use cross multiplication to eliminate the denominators:

18y * y + 12x * x = 5xy * x*y

Simplifying this expression, we get:

18y^2 + 12x^2 = 5x^2y^2

Now we can solve for x or y. Let’s solve for x:

12x^2 = 5x^2y^2 – 18y^2
7x^2 = 5x^2y^2
x^2 = 5y^2/7
x = ± (sqrt(5)/sqrt(7)) y

So the solutions to the equation 3/x + 2/y = 5/6 are x = (sqrt(5)/sqrt(7)) y and x = – (sqrt(5)/sqrt(7)) y.

In summary, cross multiplication is a useful technique for solving equations with fractions. It eliminates the need to find a common denominator and simplifies the solution process. It is especially useful in more complex equations where finding a common denominator could be time-consuming. To use cross multiplication, we simply multiply the numerator of one fraction by the denominator of the other and vice versa and then solve for the unknown variable.

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