The general form of a linear equation is of the form ax + by + c = 0 (equation of a straight line), where x and y are the variables, and a, b , c are the numerical constants. When we have pair of equations of this form we can solve them to identify if the graph of these lines are intersecting, parallel or coincide. There are various methods of solving linear equations of two variables. In this section will learn about how to solve linear equations with one or two variables, various methods, step by step solutions along with the solved examples. 
      


Steps for Solving Linear Equations

Below are few facts one should keep in mind while solving linear equations.  For any term "m"
Addition, subtraction, multiplication and division of  any number to both sides of the equation will not change the equation. In short,
  • If a = b then a + m = b + m
  • If a = b then a - m = b - m
  • If a = b then a x m = b x m
  • If a = b then a / m = b / m
Where a and b are real numbers

Mostly we deal with the linear equations contain one or two variables. Let us see how to solve linear equations step by step.

Steps to Solve a Linear Equation Involving One Unknown

Step 1: Clear the Fractions- Use the least common denominator (LCM) to clear the fractions, if given equation contains.
Step 2: Simplify both sides of the equation. 
Step 3: Arrange all the terms with the variable on one side and all constants on the other side of the equation.
Step 4: Verify your answer by substituting the values on the given equation.

How to Solve Linear Equations with Two Variable

There are various ways to solve these type of equations. 
Step 1: Clear the fraction, if any.
Step 2: Number the two given equations as (1) and (2).
Step 2: Choose any of the methods: Elimination, substitution, graphing, cross multiplication or matrix method. 
Step 3: Solve for both the variables.
Step 4: Verify your answer! 
Step 5: Write the final solution in the form (x,y).

Linear Equations Word Problems


Example: The sum of two numbers is 69 and their difference is 17. Find the numbers
Solution: Let the two numbers  are x and y.
Sum of the two numbers = x+ y = 69  -------------------(1)
Difference of two numbers = x - y = 17 ------------------(2)
Let us solve by Substitution Method.
Let us find y from Equation (1),
x + y = 69 => y = 69 - x
Substituting y = 69 - x in equation (2), we get,
 x - y = 17
=> x - ( 69 - x) = 17
=>   x - 69 +   x  = 17
=>  x + x = 17 + 69
=> 2x = 86
 => x = 86/2 = 43
Substituting x = 43 in Equation (2), we get,
x - y = 17
=>   43 - y = 17
=>  -y = 17 - 43
=> -y = - 26 
or   y = 26 [ multiplying by -1 on both sides ]
We have got x = 43 and y = 26
Hence the two numbers are 43 and 26

Examples

Solved examples on linear equations:
Example 1 : Solve 3x +  y = 8; 3x - 2y = 2

Solution: Let  3 x + y = 8 --------------------------(1)
3x - 2y = 2 --------------------------(2)

Using substitution method:
From (1), y = 8 - 3x

Substituting y = 8 - 3x in Equation (2), we get,

3x - 2 ( 8 - 3x ) = 2
=> 3x - 16 + 6x = 2


=> 9x - 16 = 2
=> 9x = 2 + 16 = 18
=>  x =  18/9 = 2
Substituting x = 2 in y = 8 - 3x, we get,
y = 8 - 6 = 2
Hence the solution is ( x, y) = ( 2, 2).

Example 2: Solve 10y + 3 = -y - 10
 
Solution: Add y to both sides
10y + 3 + y = -y - 10 + y
=> 11y + 3 = -10
Subtract 3 from both sides
11y + 3 - 3 = -10 - 3
=> 11y = -13
Divide by 11 on both sides
11y/11 = -13/11
or y = -13/11

Practice Questions

Question 1. Solve:  x + y = -1 ; y - 2x = -4

Question 2. Solve for x and y.
 4x - 5y = -4, 3x - 2y = -3

Question 3. The monthly incomes of Pinky and Kevin are in the ratio 7 : 5 and their expenditures are in the ratio 3 : 2. If each saves Rs. 1500 per month, find their monthly incomes.