Formulas To Solve Linear Equation Problems
Formulas for Linear Equations & Definitions
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.
- Read Also – Tips & tricks to solve Linear Equations question
Formulas of 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 = – b/a. We can also say that – b/a is the root of the linear equation ax + b = 0.
Formulas of 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
Formulas for 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
Solve one of the equations either for x or y.
Substitute the solution from step 1 into the other equation.
Now solve this equation for the second variable.
Multiply both the equations with such numbers to make the coefficients of one of the two unknowns numerically same.
Subtract the second equation from the first equation.
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.
Suppose there are two equation,
p1x + q1y = r1……..(1)
p2x + q2y = r2……..(2)
Multiply Equation (1) with p2
Multiply Equation (2) with p1
p₁p₂x + q₁p2y + r₁p₂ = 0
p₁p₂x + p₁q₂y + p₁r₂ = 0
q₁p₂y – p₁q₂y + r₁p₂ – r₂p₁ = 0
or, y(q₁ p₂ – q₂p₁) = r₂p₁ – r₁p₂
Therefore, y = (r₂p₁ – r₁p₂)/(q₁p₂ – q₂p₁) = (r₁p₂ – r₂p₁)/(p₁q₂ – p₂q₁) where (p₁q₂ – p₂q₁) ≠ 0
Therefore, y/(r₁p₂ – r₂p₁) = 1/(p₁q₂ – p₂q₁), …(3)
Multiply Equation (1) with q2
Multiply Equation (2) with q1
p₁q₂x + q₁q₂y + q₂r₁ = 0
p₂q₁x + q₁q₂y + q₁r₂ = 0
p₁q₂x – p₂q₁x + q₂r₁ – q₁r₂ = 0
or, x(p₁q₂ – p₂q₁) = (q₁r₂ – q₂r₁)
or, x = (q₁r₂ – q₂r₁)/(p₁q₂ – p₂q₁)
Therefore, x/(q₁r₂ – q₂r₁) = 1/(p₁q₂ – p₂q₁) where (p₁q₂ – p₂q₁) ≠ 0… (4)
From equations (3) and (4), we get,
x/(q₁r₂ – q₂r₁) = y/(r₁p₂) – r₂p₁ = 1/(p₁q₂ – p₂q₁) where (p₁q₂ – p₂q₁) ≠ 0
Note: Shortcut to solve this equation will be written as
x/(q1 r2 – q2 r1 ) = y/(r1 p2 – r2 p1 ) = (1)/(p1 q2 – p2 q1 )
x = (q1 r2– q2 r1)/(p1 q2 -p2 q1 )
y = (r1 p2 – r2 p1)/(p1 q2– p2 q1)
Important Formulas of Linear Equations & key points and Formulas to Remember
Suppose, there are two linear equations: a1x + b1y = c1 and a2x + b2y = c2
- If a1/a2= b1/b2, then there will be one solution, and the graphs will have intersecting lines.
- If a1/a2= b1/b2 = c1/c2, then there will be numerous solutions, and the graphs will have coincident lines.
- If a1/a2= b1/b2 ≠ c1/c2, then there will be no solution, and the graphs will have parallel lines.