Cross Multiplication Method

Cross multiplication is a method used to find the solution of linear equations in two variables. For a proportion like ab=cd, the cross multiplication method requires multiplying the numerator of one fraction by the denominator of the other, which results in ad = bc. This equation can then be solved step by step to find the unknown variable. 
For example, let’s solve the equation 4x=25 to find the value of x.
Cross multiply, 4 × 5 = 2 × x
20 = 2x 
x = 202=10
So, the value of x is 10.
 

For a system of two linear equations;

a₁x + b₁y = 0    (i)

a₂x + b₂y = 0    (ii)

Multiply (i) by b₂ and (ii) by b₁, then subtract,

For the 1st part of the solution, equality becomes

a₁b₂x + b₁b₂y + c₁b₂ = a₂b₁x + b₂b₁y + c₂b₁

 a₁b₂​x − a₂​b₁x + c₁​b₂ ​− c₂b₁ ​= 0

Now find x,

\( x = \frac{b_1c_2 - b_2c_1}{a_1b_2 - a_2b_1} \)

Now eliminate x to find y,

y = \( y = \frac{c_1a_2 - c_2a_1}{a_1b_2 - a_2b_1} \)

By cross multiplication, we get the formula \( x(b_1c_2 - b_2c_1) = y(c_1a_2 - c_2a_1) = 1(a_1b_2 - a_2b_1) \) 

To solve linear equations with two variables, using cross multiplication, we apply the cross multiplication formula.

For two equations; 

a₁x + b₁y + c₁ = 0

a₂x + b₂y + c₂ = 0

We use the formula x(b₁c₂-b₂c₁) = y(c₁a₂-c₂a₁)=1(a₁b₂-a₂b₁)

Now, solve for x and y using,

These are derived by the elimination method,
So,

\(x = \frac{b_1c_2 - b_2c_1}{a_1b_2 - a_2b_1} \), y=\(y = \frac{c_1a_2 - c_2a_1}{a_1b_2 - a_2b_1} \)

Let’s take an example:

2x + 3y = 5  (i)

4x + y = 11  (ii)

As the equations are not in standard form, first we rewrite them,

2x + 3y - 5 = 0   

4x + 1y - 11 = 0

Now, we apply the cross multiplication formula
\(x(b_1c_2 - b_2c_1) \)=\(y(c_1a_2 - c_2a_1) \) =1\((a_1b_2 - a_2b_1) \).
x(3)(-11) - (1)(-5) = y(-5)(4) - (-11)(2) = 1(2)(1) - (4)(3)
x - 33 + 5 = y - 20 + 22 = 12 - 12
x - 28 = y₂ = 1 - 10

Now, we solve for x and y

Solve for x

x - 28 = 1 - 10 - 10x = - 28x = -28 - 10 = 2.8

Solve for y

y₂ = 1 -1 0 - 10y = 2x = 2 - 10 = 0.2

We now know that x = 2.8 and y = 0.2.

To cross-multiply fractions, we must multiply the denominator of one fraction by the numerator of the other fraction and then compare the products. So, for fractions, ab=cd we cross-multiply ad=bc.
For example;

23 = 46

2 × 6 = 12

4 × 3 = 12

Since both the products are equal, 2 × 6 = 4 × 3 = 12. So the fractions are equivalent.
 

To cross-multiply three fractions, all the numerators are multiplied by each other, and all the denominators are also multiplied. Then we get a new fraction as the answer. We can simplify it if needed. In mathematical terms, for 3 fractions abcdef, we get the result a  c  eb  d  f.

For example,

234512=2  4  13  5  2=830

We can simplify it further, 830=415

Now, divide the numerator and denominator by 2.

8/230/2= 415

So, cross-multiplying 234512 that gives us 415

We can use cross multiplication to compare two fractions and determine which one is larger, smaller, or equivalent. To compare ab and cd, we multiply ad and cb. 

If ad>bc, then ab>cd

If ad<bc, then ab<cd

For example, let’s compare the fractions 34 = 23

3 × 3 = 9

2 × 4 = 8

9 > 8, so 34 > 23
 

Cross multiplication is useful in comparing ratios and finding values. Similar to comparing fractions, the numerator of the first ratio is multiplied by the denominator of the second, and the denominator of the first is multiplied by the numerator of the second ratio. So, two ratios a:b and c:d written as ab and cd can be compared by cross multiplying ad and bc.
For example, let’s compare 7:9 and 5:6
Let’s write the ratios as fractions, 79 and 56

7 × 6 =  42

9 × 5 = 45

42 < 45, so 7:9 < 5:6

The cross multiplication method is commonly used when solving proportions.

For xa=bc, 

xc=ab 

so x=abc.

Let us take an example to understand better.

x₄=35

x₅=43

5x=12

x=125

x=2.4

Cross multiplication involving variables on both sides has the same method of multiplication as with a single variable. Let us take an example to see how.

Example: x + 23 = 2x - 15

Cross multiply, (x + 1)5 = (2x - 1)3

Expand both sides, 5x + 10 = 6x - 3

Now, we solve for x

10 + 3 = 6x - 5x

x = 13

The cross multiplication method is a quick and systematic way to solve two-variable linear equations. It replaces lengthy elimination steps with simple diagonal multiplication and subtraction for easy results.
 

• The cross multiplication method is used to solve linear equations in two variables. It involves multiplying the coefficients diagonally and subtracting the products.
   

• This method simplifies solving equations without using substitution or elimination. It gives direct formulas for finding the values of x and y
   

• To apply it, draw a cross between the coefficients of x, y and constants. Then multiply diagonally and take their difference for each variable.
   

• Always check the signs carefully during multiplication. A small sign error can completely change your final answer.
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• Use neat and clear alignment when writing equations. This prevents confusion between coefficients and constants.

Cross multiplication method is useful for solving problems that involve ratios, and it is used in real-world tasks requiring proportional comparisons or unit conversions. Some such real-life uses of the cross multiplication method are as follows:

Adjusting recipes while cooking: When a recipe needs to be cooked for more or fewer people than suggested, cross-multiplication is useful for maintaining the ingredient ratios. For instance, if a recipe requires 3 cups of flour for 3 servings, then we can find how many would be required for 5 servings.

Scaling blueprints in architecture: Architects and engineers use a cross multiplication method to scale a drawing to fit on paper, this method helps maintain accurate proportions. For example, If the 2cm on the blueprints represents 5 meters in real life, this cross multiplication method can be used to find how many centimeters represent 20 meters.

Map reading and scale calculations: Cross multiplication helps determine actual distances using map scales. For instance, on a map, 1cm = 4km, the real distance between two points 7cm apart on the map can be found using cross multiplication. 

Comparing prices: Cross multiplication helps compare costs per unit, which helps compare prices and calculate discounts while shopping.

Solving mixture problems in chemistry: When dealing with mixtures and ratio problems in Chemistry, cross-multiplication helps provide quick solutions. For example, if a solution has salt and water in the ratio of 1:5, then how much salt is required for 15 liters of water?