Formulas and Methods to solve Linear equations
Step 1: Solve one of the equations either for x or y.
Step 2: Substitute the solution from step 1 into the other equation.
Step 3: Now solve this equation for the second variable.
Step 1: Multiply both the equations with such numbers to make the coefficients of one of the two unknowns numerically same.
Step 2: Subtract the second equation from the first equation.
Step 3: In either of the two equations, substitute the value of the unknown variable. So, by solving the equation, the value of the other unknown variable is obtained.
- Cross-Multiplication Method
Suppose there are two equation,
p_{1}x +q_{1}y = r_{1} ……..(1)
p_{2}x +q_{2}y = r_{2} ……..(2)
Multiply Equation (1) with p2
Multiply Equation (2) with p1
p_{1}p_{2}x +q_{1}p_{2}y = r_{1}p_{2}
p_{1}p_{2}x +p_{1}q_{2}y = p_{1}r_{2}
Subtracting,
q_{1}p_{2}y – p_{1}q_{2}y = r_{1}p_{2} – p_{1}r_{2}
or, y (q₁ p₂ – q₂p₁) = r₂p₁ – r₁p₂
Therefore, y = \frac{r_{2}p_{1} – r_{1}p_{2} }{q_{1}p_{2} – q_{2}p_{1} }
= \frac{r_{1}p_{2} – r_{2}p_{1} }{q_{2}p_{1} – q_{1}p_{2} }
where (p₁q₂ – p₂q₁) ≠ 0
Therefore, \frac{y}{r_{1}p_{2} – r_{2}p_{1} } = \frac{1}{q_{2}p_{1} – q_{1}p_{2} } …(3)
Multiply Equation (1) with q2
Multiply Equation (2) with q1
p_{1}q_{2}x +q_{1}q_{2}y = r_{1}q_{2}
q_{1}p_{2}x +q_{1}q_{2}y = q_{1}r_{2}
Subtracting,
p_{1}q_{2}x – p_{2}q_{1}x = r_{1}q_{2} – q_{1}r_{2}
or , x(p₁q₂ – p₂q₁) = (q₁r₂ – q₂r₁)
or, x = \frac{q_{1}r_{2} – r_{1}q_{2}}{p_{1}q_{2} – p_{2}q_{1}}
Therefore, \frac{x}{q_{1}r_{2} – r_{1}q_{2}} = \frac{1}{p_{1}q_{2} – p_{2}q_{1}} … (4)
where (p₁q₂ – p₂q₁) ≠ 0
From equations (3) and (4), we get,
\frac{x}{q_{1}r_{2} – r_{1}q_{2}} =\frac{y}{r_{1}p_{2} – p_{1}r_{2}} = \frac{1}{p_{1}q_{2} – p_{2}q_{1}}
where (p₁q₂ – p₂q₁) ≠ 0
Note: Shortcut to solve this equation will be written as
\frac{x}{q_{1}r_{2} – r_{1}q_{2}} =\frac{y}{r_{1}p_{2} – r_{2}p_{1} } = \frac{1}{q_{2}p_{1} – q_{1}p_{2} }
which means,
x = \frac{q_{1}r_{2} – r_{1}q_{2}}{q_{2}p_{1} – q_{1}p_{2} }
y = \frac{r_{1}p_{2} – r_{2}p_{1}}{q_{2}p_{1} – q_{1}p_{2}}
Login/Signup to comment