**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 p_{2 }

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 q_{2 }

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}}

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