# Linear Equations Solver

Many students who experienced a staggering blow with the complicated linear equations specially system of linear equations with fractions and it has been very difficult for them to overcome. That is the reason, we came up with some simple tricks to solve system of equations where students feel confident and solve questions at their own pace. Use of linear equations solver online is very simple, just insert your equations and click on submit button. Your answer will be ready in few seconds. In this section will help you to understand how to solve equations either with one unknown or with two unknowns.

## Solving Linear Equation in One Variable

The linear equation of one variable is of the form ax + b = 0, where a and b are real numbers and a is not equal to 0. This equation has one and only solution, the value of the variable x is unique i.e. x = - $\frac{b}{a}$.

Solving two step linear equations
For any given linear equation of the form ax + b = 0,
Step 1: Subtracting b from both sides,
ax + b - b = 0 - b [ since b - b = 0 ]
=> ax = -b
Step 2: Dividing by a on both sides.
=> $\frac{ax}{a}$ =  $\frac{-b}{a}$
=> x = -$\frac{b}{a}$
Hence the final solution is x = - b/a

For Example: Solve 3x +  12 = 0
Subtract 12 from both the sides
3x + 12 - 12 = 0 -12
3x = -12
To isolate x, divide both the sides by 3
3x/3 = -12/3

## Solving System of Linear Equation in Two Variables

General form for the equation is ax + by + c = 0, where a and b are non zero real numbers and c is any constant. To find the value of two given unknowns, we need two equations containing same unknowns with degree one.
Consider two linear equations with two unknowns x and y be,
$a_1 x + b_1 y +c_1$ =  0
$a_2 x + b_2 y + c_2$ = 0

Solutions of system of linear equations of two variables
1. Unique solution:  When the two lines intersect the, the point of intersection is the common solution to the two equations, since it satisfy both the equations.
2. No solution: When the two lines do not intersect, they will be parallel to each other.  Hence the pair of equations do not have any solution.
3. Infinite number of solutions:  When the two lines coincide, there will be infinite number of points common to both the lines when we graph them. Hence there will be infinite number of common solutions.

## Solving Linear Equations without Calculator

Below we have illustrated few of the problems using different methods:
Example 1 : Find the point(s) to the given equations : 2x + y = 7 and 3x - y =3
Solution: Let us solve the equations by Elimination Method.
Let us first number the equations as (1) and (2)
2x + y = 7 -------------------------------(1)
3 x -  y = 3 -------------------------------(2)

On adding (1) and (2), we get
5x   = 10  [ since + y - y = 0 ]
x = 10/5
x = 2
Substituting x=2, in Equation (1), we get,
2(2) + y = 7
4 + y  = 7
y  = 7 - 4 = 3
=>   y  = 3
Hence the common solution, which is the point of intersection of the two lines represented by the given two equations is (x,y) = ( 2, 3).

Example 2: Solve : 2x + 3y = 11 ; x - 2y = 2 by the Method of Substitution.
Solution: By the method of substitution we find one of the variable from one of the equation and substitute it in the other equation.
We have, 2x - 3 y = 11 -------------------(1)
and x - 2y = 2   -------------------(2)
x - 2y = 2
=>  x = 2 + 2y
Substituting x = 2 - 2y in equation (1), we get
2 ( 2 + 2y ) + 3y = 11
=>4 + 4y + 3y = 11
=>  4 + 7y = 11
=>7y = 11 - 4 = 7
=>y = 7/7 = 1
Substituting
y = 1, in x = 2 + 2y, we get
x = 2 + 2(1) = 2 + 2= 4
Hence the solution to the above equation is ( x, y ) = ( 4, 1)

Example 3: Find the solution to both the equations: x + y = 4 and 2x + 3y = 9. Represent by graphical method.
Solution: To find the solution of given set of equations graphically, follow below steps:
1. Graph both the equations in same coordinate system.
2. Record intersection ordered pair. Which is the solution.

Graphs of  x + y = 4 (red line) and 2x + 3y = 9 (blue line)

Therefore, (3,1) is the solution to the system since both the lines intersect at the point.
Verification: x + y = 4, Put x = 3 and y = 1
3 +1 = 4
4=4(True)

## Practice Questions

1. Solve the following linear equations of one variable by two step method.
a.  4x + 15 = -1
b.  5 + 3x = 11
2. Solve the following pairs of equations, write the common solution/solutions if any.
a.  3x + 5y = -2 ; 4x - y = 5
b.  4x + 3y = 10, 12 x + 9 y = 30
c.  8x + 3y = 11, 16 x + 6y = 22