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Basic Concept of Linear Equations
A linear equation is also known as an algebraic equation in which each term has an exponent of one. The graph representation of the equation shows a straight line. Standard form of linear equation is y = m x + b. Where, x is the variable and y, m, and b are the constants. On this page you will find complete Formulas for Linear Equation that will help you to solve questions of all levels.


Formulas & Definitions for Linear Equations
- A linear equation is an algebraic equation in which each term has an exponent of one and the graphing of the equation results in a straight line.
- Standard form of linear equation is y = mx + b. Where, x is the variable and y, m, and b are the constants.
Linear equations in one variable
- A Linear Equation in one variable is defined as ax + b = 0
- Where, a and b are constant, a ≠ 0, and x is an unknown variable
- The solution of the equation ax + b = 0 is x = – \frac{b}{a} . We can also say that – \frac{b}{a} is the root of the linear equation ax + b = 0.
Linear equations in two variable
- A Linear Equation in two variables is defined as ax + by + c = 0
- Where a, b, and c are constants and also, both a and b ≠ 0
Linear equations in three variable
- A Linear Equation in three variables is defined as ax + by + cz = d
- Where a, b, c, and d are constants and also, a, b and c ≠ 0
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}}
Important Formulas of Linear Equation & key points to Remember
- Suppose, there are two linear equations: a1x + b1y = c1 and a2x + b2y = c2
Now,
(A) If \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}, then there will be one solution, and the graphs will have intersecting lines.
(B) If\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}, then there will be numerous solutions, and the graphs will have coincident lines.
(C) If \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}≠ \frac{c_{1}}{c_{2}} then there will be no solution, and the graphs will have parallel lines
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