Cross Multiplication- Pair Of Linear Equations In Two Variables
Cross multiplication is a method for solving simultaneous linear equations in two variables. It is crucial because it helps find the values of unknown variables in a system of linear equations when traditional methods like elimination or substitution are not applicable.
Intoduction
The cross multiplication method involves multiplying the coefficients of each equation in a system by the coefficients of the other equation, then equating the products. This creates a new equation that can be solved for one variable, after which the other variable can be determined using the original equation.
Cross-multiplication Definition
The cross-multiplication method is a method used to solve a system of linear equations by multiplying the coefficients of one equation by the constants of the other equation and then equating the products.
Consider a system of two linear equations:
a1x+b1y+c1=0 and a2x+b2y+c2=0
The cross-multiplication method involves multiplying the coefficients of one equation by the constants of the other equation, such that the variables x and y are eliminated. After the elimination of x and y, we get
x=b1c2 - b2c1b2a1 - b1a2 y=c1a2 - c2a1b2a1 - b1a2
Where, b2a1 - b1a2≠0
Also Read: Degree of Polynomial
Formula for cross-multiplication
We use the cross-multiplication formula to solve linear equations involving two variables:
xb1c2 - b2c1 = -ya1c2 - a2c1 = 1a1b2 - a2b1
Derivation of cross-multiplication
The derivation of the cross multiplication method can be traced back to the concept of proportional reasoning. If two ratios are equal, then their cross-products must also be equal. This concept can be applied to simultaneous linear equations in two variables to find their solutions.
In general, a pair of linear equations in two variables can be represented as
a1x+b1y+c1=0 and a2x+b2y+c2=0
The steps below are used to solve a pair of linear equations in two variables:
assuming two linear equations with two variables
a1x+b1y+c1=0…(1) a2x+b2y+c2=0…(2)
Step i) Equation (1) is multiplied by b2 and Equation (2) by b1.
b2a1x+b2b1y+b2c1=0…(3) b1a2x+b1b2y+b1c2=0…(4)
Step ii) Subtracting Equation (4) from (3):
b2a1-b1a2x+b2b1-b1b2y+b2c1-b1c2=0 b2a1-b1a2x=b1c2-b2c1
x=b1c2-b2c1b2a1-b1a2, given b2a1-b1a2≠0
Also Read: Cross Section
Once x is found, it can be substituted into either of the original equations to find the value of y.
Step iii) After substituting the value of x in any of the equations, the value of " y " obtained is:
y=c1a2-c2a1b2a1-b1a2
The solutions are given as
xb1c2-b2c1=yc1a2-c2a1=1b2a1-b1a2
This method can also be applied to systems with more than two equations by performing cross multiplications with each pair of equations, and successively eliminating variables until a unique solution is found.
This method is known as the "Cross-Multiplication Method" because it can be used to simplify the solution and, as a result, aid with memorization. The following diagram can be used to help you remember the cross-multiplication technique for solving linear equations with two variables:
Read More: Cube and Cuboid
Uses of cross-multiplication
Cross multiplication is useful in mathematics for solving problems involving proportions, ratios, and rates. It can also be used to solve systems of linear equations with more than two variables by reducing them to smaller systems with two variables.
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Frequently Asked Questions on Cross Multiplication- Pair Of Linear Equations In Two Variables
Ans. Cross multiplication is a technique used to solve equations involving fractions by multiplying across the diagonals.
Ans. Cross multiplication works by multiplying the numerator of one fraction by the denominator of the other, and setting the products equal to each other.
Ans. The rules for cross multiplying include multiplying across the diagonals and equating the products, ensuring the fractions are set equal before starting.
Ans. The method of cross multiplication involves clearing the fractions by multiplying each numerator by the opposite denominator and solving the resulting equation.