Class X – Maths Formula – Pair of Linear Equations in Two Variables

Linear Equations

One Variable	a𝑥 + b = 0	a ≠ 0 and a & b are real numbers
Two Variable	a𝑥 + b𝑦 + c = 0	a ≠ 0, b ≠ 0 and a, b & c are real numbers
Three Variable	a𝑥 + b𝑦 + cz + d = 0	a ≠ 0, b ≠ 0, c ≠ 0 and a, b, c & d are real numbers

Pair of Linear Equations in two Variables

a1𝑥 + b1𝑦 + c1 = 0 a2𝑥 + b2𝑦 + c2 = 0

Where a1, b1, c1, a2, b2 and c2 are all real numbers and a12 + b12 ≠ 0 & a22 + b22 ≠ 0

Important Points

1. A linear equation in two variable has infinite solutions.

2. The graph of every linear equation in two variable is a straight line.

3. 𝑥 = 0 is the equation of the 𝑦-axis and 𝑦 = 0 is the equation of the 𝑥-axis

4. The graph 𝑥 = a is a line parallel to 𝑦-axis

5. The graph 𝑦 = b is a line parallel to 𝑥-axis

6. An equation of the type 𝑦 = m𝑥 represents a line passing through the origin.

Linear equation in one Variable	Solution: One
Linear equation in two Variable	Solution: Infinite solution possible
Linear equation in three Variable	Solution: Infinite solution possible

a1𝑥 + b1𝑦 + c1 = 0 a2𝑥 + b2𝑦 + c2 = 0	a1/ a2 ≠ b1/ b2	Intersecting lines	One unique solution only, consistent
a1𝑥 + b1𝑦 + c1 = 0 a2𝑥 + b2𝑦 + c2 = 0	a1/ a2 = b1/ b2 = c1/ c2	Coincident lines	Infinite solutions, consistent
a1𝑥 + b1𝑦 + c1 = 0 a2𝑥 + b2𝑦 + c2 = 0	a1/ a2 = b1/ b2 ≠ c1/ c2	Parallel lines	No solution, inconsistent

1. Method of elimination by substitution	Steps: 1) Suppose the equation are a1𝑥 + b1𝑦 + c1 = 0 a2𝑥 + b2𝑦 + c2 = 0 2. Find the value of variable of either 𝑥 or 𝑦 in other variable term in first equation. 3. Substitute the value of that variable in second equation. 4. Now this is a linear equation in one variable. Find the value of the variable. 5. Substitute this value in first equation and get the second variable.
2. Method of elimination by equating the coefficients	Steps: 1) Suppose the equation are a1𝑥 + b1𝑦 + c1 = 0 a2𝑥 + b2𝑦 + c2 = 0 2. Find the LCM of a1 and a2. Let it be k 3. Multiply the first equation by the value k/a1 4. Multiply the first equation by the value k/a2 5. Subtract the equation obtained. This way one variable will be eliminated and we can solve to get the value of variable in 𝑦. 6. Substitute this value in first equation and get the second variable.
3. Cross Multiplication method	Steps: 1) Suppose the equation are a1𝑥 + b1𝑦 + c1 = 0 a2𝑥 + b2𝑦 + c2 = 0 2. This can be written as 3. 4. 𝑥 => first and last expression 𝑦 => second and last expression