Here we will discuss about simultaneous linear equations by using crossmultiplication method.
General form of a linear equation in two unknown quantities:
ax + by + c = 0, (a, b ≠ 0)
Two such equations can be written as:
a₁x + b₁y + c₁ = 0  (i)
a₂x + b₂y + c₂ = 0  (ii)
Let us solve the two equations by the method of elimination, multiplying both sides of equation (i) by a₂ and both sides of equation (ii) by a₁, we get:
a₁a₂x + b₁a₂y + c₁a₂ = 0
a₁ a₂x + a₁b₂y + a₁c₂ = 0
Subtracting, b₁a₂y  a₁b₂y + c₁a₂  c₂a₁ = 0
or, y(b₁ a₂  b₂a₁) = c₂a₁  c₁a₂
Therefore, y = (c₂a₁  c₁a₂)/(b₁a₂  b₂a₁) = (c₁a₂  c₂a₁)/(a₁b₂  a₂b₁) where (a₁b₂  a₂b₁) ≠ 0
Therefore, y/(c₁a₂  c₂a₁) = 1/(a₁b₂  a₂b₁),  (iii)
Again, multiplying both sides of (i) and (ii) by b₂ and b₁ respectively, we get;
a₁b₂x + b₁b₂y + b₂c₁ = 0
a₂b₁x + b₁b₂y + b₁c₂ = 0
Subtracting, a₁b₂x  a₂b₁x + b₂c₁  b₁c₂ = 0
or, x(a₁b₂  a₂b₁) = (b₁c₂  b₂c₁)
or, x = (b₁c₂  b₂c₁)/(a₁b₂  a₂b₁)
Therefore, x/(b₁c₂  b₂c₁) = 1/(a₁b₂  a₂b₁) where (a₁b₂  a₂b₁) ≠ 0  (iv)
From equations (iii) and (iv), we get:
x/(b₁c₂  b₂c₁) = y/(c₁a₂)  c₂a₁ = 1/(a₁b₂  a₂b₁) where (a₁b₂  a₂b₁) ≠ 0
This relation informs us how the solution of the simultaneous equations, coefficient x, y and the constant terms in the equations are interrelated, we can take this relation as a formula and use it to solve any two simultaneous equations. Avoiding the general steps of elimination, we can solve the two simultaneous equations directly.
So, the formula for crossmultiplication and its use in solving two simultaneous equations can be presented as:
If (a₁b₂  a₂b₁) ≠ 0 from the two simultaneous linear equations
a₁x + b₁y + c₁ = 0  (i)
a₂x + b₂y + c₂ = 0  (ii)
we get, by the crossmultiplication method:
x/(b₁c₂  b₂c₁) = y/(c₁a₂  c₂a₁) = 1/(a₁b₂  a₂b₁)  (A)
That means, x = (b₁c₂  b₂c₁)/(a₁b₂  a₂b₁)
y = (c₁a₂  c₂a₁)/(a₁b₂  a₂b₁)
Note:
If the value of x or y is zero, that is, (b₁c₂  b₂c₁) = 0 or (c₁a₂  c₂a₁) = 0, it is not proper to express in the formula for cross multiplication, because the denominator of a fraction can never be 0.
From the two simultaneous equations, it appears that the formation of relation (A) by crossmultiplication is the most important concept.
At first, express the coefficient of the two equations as in the following form:
Now multiply the coefficient according to the arrow heads and subtract the upward product from the downward product. Place the three differences under x, y and 1 respectively forming three fractions; connect them by two signs of equality.
Workedout examples on simultaneous linear equations by using crossmultiplication method:
1. Solve the two variables linear equation:
8x + 5y = 11
3x – 4y = 10
Solution:
On transposition, we get
8x + 5y – 11 = 0
3x – 4y – 10 = 0
Writing the coefficient in the following way, we get:
Note: The above presentation is not compulsory for solving.
By crossmultiplication method:
x/(5) (10) – (4) (11) = y/(11) (3) – (10) (8) = 1/(8) (4) – (3) (5)
or, x/50 – 44 = y/33 + 80 = 1/32 – 15
or, x/94 = y/47 = 1/47
or, x/2 = y/1 = 1/1 [multiplying by 47]
or, x = 2/1 = 2 and y = 1/1 = 1
Therefore, required solution is x = 2, y = 1
2. Find the value of x and y by using the using crossmultiplication method:
3x + 4y – 17 = 0
4x – 3y – 6 = 0
Solution:
Two given equations are:
3x + 4y – 17 = 0
4x – 3y – 6 = 0
By crossmultiplication, we get:
x/(4) (6) – (3) (17) = y/(17) (4) – (6) (3) = 1/(3) (3) – (4) (4)
or, x/(24 – 51) = y/(68 + 18) = 1/(9 – 16)
or, x/75 = y/50 = 1/25
or, x/3 = y/2 = 1 (multiplying by 25)
or, x = 3, y = 2
Therefore, required solution: x = 3, y = 2.
3. Solve the system of linear equations:
ax + by – c² = 0
a²x + b²y – c² = 0
Solution:
x/(b + b²) = y/( a² + a) = c²/(ab²  a²b)
or, x/b(1  b) = y/ a(a  1) = c²/ab(a  b)
or, x/b(1  b) = y/a(a  1) = c²/ab(a  b)
or, x = bc²(1 – b)/ab(a – b) = c²(1 – b)/a(a – b) and y = c²a(a – 1)/ab(a – b) = c²(a – 1)/b(a – b)
Hence the required solution is:
x = c²(1 – b)/a(a – b)
y = c²a(a – 1)/b(a – b)
● Simultaneous Linear Equations
Solvability of Linear Simultaneous Equations
Word Problems on Simultaneous Linear Equations
Word Problems on Simultaneous Linear Equations
Practice Test on Word Problems Involving Simultaneous Linear Equations
● Simultaneous Linear Equations  Worksheets
Worksheet on Simultaneous Linear Equations
Worksheet on Problems on Simultaneous Linear Equations
8th Grade Math Practice
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