Formulas for Linear Equation Problems

Formulas and Methods to solve Linear equations

• Substitution Method

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.

• Elimination Method

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}}