Cross-multiplication is a mathematical method used to solve equations involving two equal fractions or ratios. It provides a quick and efficient way to solve proportional relationships without first finding a common denominator.
- The numerator of the first fraction is multiplied by the denominator of the second fraction.
- The denominator of the first fraction is multiplied by the numerator of the second fraction.
- If the resulting products are equal, the proportion is true.
If the given pair of linear equations is,
- (a)1x + (b)1y + (c)1 = 0
- (a) 2x + (b) 2y + (c) 2 = 0
By using cross multiplication, the values of x and y will be
x=\frac{b_1c_2-b_2c_1}{b_2a_1-b_1a_2}\\[3pts]y=\frac{c_1a_2-c_2a_1}{b_2a_1-b_1a_2} where b2a1 - b1a2 ≠ 0
The required cross-multiplication formula is,
\frac{x}{b_1c_2 - b_2c_1} = \frac{-y}{a_1c_2-a_2c_1 } = \frac{1}{b_2a_1 - b_1a_2}
Derivation
We can easily derive the cross-multiplication method formula for a pair of linear equations in two variables by simply solving the linear equations by making their coefficients equal.
Let's take two linear equations as
(a)1x + (b)1y + (c)1 = 0...(i)
(a)2x + (b)2y + (c)2 = 0...(ii)
To solve the above pair of linear equations, we make the coefficient of y in both equations equal.
Multiplying b2 to eq(i) and b1 to eq(ii)
(b2)(a)1x + (b2)(b)1y + (b2)(c)1 = 0...(iii)
(b1)(a)2x + (b1)(b)2y + (b1)(c)2 = 0...(iv)
Subtracting equations (iii) and (iv) we get
(b2a1 - b1a2)x + (b2c1 - b1c2) = 0
Solving for x we get, x = (b1c2 - b2c1)/(b2a1 - b1a2).........(1) where, (b2a1 - b1a2) ≠ 0
Similarly solving eq (i) and eq (ii) for y we get, y = (c1a2 - c2a1)/(b2a1 - b1a2)...........(2) where, (b2a1 - b1a2) ≠ 0
Combining eq.(1) and eq.(2) we get,
\frac{x}{b_1c_2 - b_2c_1} = \frac{y}{c_1a_2 - c_2a_1} = \frac{1}{b_2a_1 - b_1a_2} This is the required cross-multiplication formula.
Conditions for Solutions of Linear Equations
Unique Solution: We get a unique solution to the pair of linear equations if the given pair of lines is consistent. For any two lines a₁x + b₁y = -c₁ and a₂x + b₂y = -c₂ to be consistent, the required condition is,
\frac{a_1}{a_2}\neq\frac{b_1}{b_2} ora_1b_1 - a_2b_2 = 0
The line for two equations will intersect at a unique point, which is the solution of the two equations.
No solution (Inconsistent Solution): We can have no solution to the pair of linear equations if the given pair of lines are parallel but their y-intercepts are not equal. For any two lines a₁x + b₁y = -c₁ and a₂x + b₂y = -c₂ to have no solution, the required condition is,
\frac{a_1}{a_2}=\frac{b_1}{b_2} \neq \frac{c_1}{c_2} ora_1b_2-a_2b_1=0 ora_1c_2-a_2c_1\neq0
In case there is no solution, the graphs of two equations will be parallel, and hence no solution is derived.
Infinitely Many Solutions: We can have Infinitely Many solutions to the pair of linear equations if the given pair of lines are parallel but their y-intercepts are also multiples of each other. For any two lines a₁x + b₁y = -c₁ and a₂x + b₂y = -c₂ to have infinite solutions, the required condition is,
\frac{a_1}{a_2}=\frac{b_1}{b_2} = \frac{c_1}{c_2} or
a1b2 - a2b1 = 0 , b1c2 - b2c1 = 0 , a1c2 - a2c1 = 0
In the case of Infinitely many solutions, the two graphs will be overlapping each other, thus giving infinitely many solutions.
