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.

Linear equations formula

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

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,

p1x + q1y = r1……..(1)

p2x + q2y = r2……..(2)

Multiply Equation (1) with p

Multiply Equation (2) with p1

p₁p₂x + q₁p2y + r₁p₂ = 0

p₁p₂x + p₁q₂y + p₁r₂ = 0

Subtracting,
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 q

Multiply Equation (2) with q1

p₁q₂x + q₁q₂y + q₂r₁ = 0

p₂q₁x + q₁q₂y + q₁r₂ = 0

Subtracting, 

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 )

which means,

x = (q1 r2– q2 r1)/(p1 q2 -p2 q1 )

y = (r1 p2 – r2 p1)/(p1 q2– p2 q1)

Formulas for Linear Equations

Important Formulas of Linear Equations & key points and Formulas to Remember

Suppose, there are two linear equations: a1x + b1y = c1 and a2x + b2y = c2

Now,

  • 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.

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