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In Maths, we use the cross multiplication method for solving linear equations in two variables. This is the simplest method to solve them and gives us the accurate value of the two variables. However, the method of cross multiplication is applicable only when we have a pair of linear equations in two variables.

Consider that a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are two equations that need to be solved. By the method of cross-multiplication, we would find the values of the x and y variables.

x = \[\frac{b_{1}c_{2}-b_{2}c_{1}}{b_{2}a_{1}-b_{1}a_{2}}\] and y = \[\frac{c_{1}a_{2}-c_{2}a_{1}}{b_{2}a_{1}-b_{1}a_{2}}\]

Here, b2a1 - b1a2 ≠ 0

The final solution is,

\[\frac{x}{b_{1}c_{2}-b_{2}c_{1}}\] = \[\frac{y}{c_{1}a_{2}-c_{2}a_{1}}\] = \[\frac{1}{b_{2}a_{1}-b_{1}a_{2}}\]

This method given above is known as the ‘Cross-Multiplication Method’ as this technique is much useful when it comes to simplifying the solution. For memorizing the method of cross-multiplication and solving the linear equation in two variables, the diagram given below is helpful.

(Image to be added soon)

The arrows in the diagram indicate the multiplication of the values that are connected through the arrow. We can see that the second product is then subtracted from the first one. The result that we get is substituted as the denominator of the given variables and 1, as we can see above the arrow. The entire values that are obtained are then equated to form an equation as follows.

\[\frac{x}{b_{1}c_{2}-b_{2}c_{1}}\] = \[\frac{y}{c_{1}a_{2}-c_{2}a_{1}}\] = \[\frac{1}{b_{2}a_{1}-b_{1}a_{2}}\]

From here, we can evaluate the values of x and y, provided that b2a1 - b1a2 ≠ 0. Hence, we call this method cross-multiplication.

In this method, the condition for the consistency of the given pair of linear equation in two variables has to be checked. These conditions are as follows:

If \[\frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}\], the result is a unique solution and the given pair of the linear equations in two variables are known to be consistent.

If \[\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}\] , there would be infinite solutions and the pair of the lines would be coincident and hence, consistent and dependent.

If \[\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}\], then there would be no existing solution and the pair of the given linear equations in two variables are inconsistent.

Let us consider the following examples.

Example 1: Solve the following pair of linear equations by using the cross multiplication method.

3x − 4y = 2

y − 2x = 7

Solution:

We can rewrite the above-given equation as:

3x − 4y = 2

−2x + y = 7

By the method of the cross multiplication,

\[\frac{x}{b_{1}c_{2}-b_{2}c_{1}}\] = \[\frac{y}{c_{1}a_{2}-c_{2}a_{1}}\] = \[\frac{1}{b_{2}a_{1}-b_{1}a_{2}}\]

When we substitute the values from the above given equation,

\[\frac{x}{28+2}\] = \[\frac{y}{4+21}\] = \[\frac{1}{13-8}\]

\[\frac{x}{30}\] = \[\frac{x}{25}\] = -15

Hence, x = -6 and y = -5

Example 2: Determine the value of the given variables that satisfy the following equation:

2x + 5y = 20 and 3x+6y =12.

Solution:

We have,

2x+5y = 20

3x + 6y = 12

By the cross multiplication method, we know that

Hence, by substituting the values that we have in the above equation, we get,

\[\frac{x}{ [(5).(12) - (6).(20)] }\] = \[\frac{y}{ [(20).(3) - (12).(2)] }\] = \[\frac{1}{ [(5).(3) - (6).(2)] }\]

\[\frac{x}{(60-120)}\] = \[\frac{y}{(60-24)}\] = \[\frac{1}{(15-12)}\]

\[\frac{x}{(-60)}\] = \[\frac{y}{36}\] = \[\frac{1}{3}\]

On solving, we get,

\[\frac{x}{-60}\] = \[\frac{1}{3}\]

Hence, x = -20

And solving for y,

\[\frac{y}{36}\] = \[\frac{1}{3}\]

Hence, y = 12

Therefore, we have, x= -20 and y = 12, which is the point at which the given two equations would intersect.

FAQ (Frequently Asked Questions)

1. What do you mean by the cross-multiplication method?

The cross-multiplication method is explained as follows.

In the elementary algebra and elementary arithmetic, when there is an equation given between two fractions or two irrational numbers, we can cross-multiply the two for either simplifying the equation or finding out the values of the unknown variables. This method is referred to as the method of cross-multiplication. Practically, this method of cross-multiplication means that we need to multiply the numerator of one or each side of the given fraction by the denominator of the other side of the fraction, in a way that the terms tend to cross over.

2. What is the Rule of 3 Method in mathematics?

The rule of 3 in terms of mathematics is an operation which helps us in quickly solving both the direct and the inverse proportion word problems. If we have to use the rule of 3, we must have three values: two values which are proportional to each other and a third value. From here, we can determine the value of the fourth one.

In simpler terms, the rule of three is a mathematical method in which we can determine the solution depending on the proportions. The best example of this is the cross-multiplication method wherein we find the values of the unknown variables by writing them in a proportion.