What are Equations?

Equations are those in which the algebraic expressions are equated to a constant or an algebraic expression. For example, 2x + 3y = 5z. The Equations are solved to find the exact value of the variable involved in the equation.

**The general form of a linear equation with**,

- One variable is

**ax + b = 0,**where a ≠ 0 and x is the variable.

- Two variables is a

**x + by + c = 0**, where, a ≠ 0, b ≠ 0 , x and y are the variables.

- Three variables is

**ax + by + cz + d = 0**where a ≠ 0, b ≠ 0, c ≠ 0, x, y, z are the variables.

### How to solve linear Equations?

For the equations involving 2 variables, we need to have two equations to solve for the two variables. There are various methods to solve these equations. Some of them are as follows:- Cross multiplication method
- Method of substitution
- Method of elimination
- Matrix method
- Determinant methods

### Solved Example

**Question:**Solve x + y = 5

**Solution:**

=> y = 5 - x

when x = -3, y = 5 - (-3) = 5 + 3 = 8 , the solution is ( -3, 8)

when x = 0, y = 5 - 0 = 5 , the solution is ( 0, 5)

When x = 5, y = 5 + 5 = 10, the solution is ( 5, 0)

x |
-3 |
0 |
5 |

y | 8 | 5 |
0 |

The table contains some of the solutions on the above linear equation. We can also find the values of y for the different values of x.

- Graph of linear equations of one variable is a point on the real number line.

**For Example**: Draw graph for the linear equation, 2x + 5 = 9

Given 2x + 5 = 9

=> 2x + 5 - 5 = 9 - 5

=> 2x = 4

=> $\frac{2x}{2} = \frac{4}{2}$

=> x = 2

This can be represented on the real number line as follows:

In the above number line we can see that the solution, x = 2 is marked.

- Graph of linear Equations of 2 variables will be a straight line, which can be shown on co-ordinate graph.

We have the points, (-3, 8), (0, 5), (5, 0)

System of linear equations are of the form:

a

_{1}x

_{ }+ b

_{1}y + c

_{1}= 0 and

a

_{2}x

_{ }+ b

_{2}y + c

_{2}= 0

These system of equations have following types of solutions according to the ratio of the corresponding coefficients. The above system of equations may have unique solution or no solution or infinite number of solutions.

Linear Equations of 2 or more variables can be solved by various methods as discussed in the paragraph "linear equations solver".

Let us discuss the method of substitution:

### Solved Example

**Question:**Solve the following equations by the method of substitution.

x + y = 12,

2x + 3y =32

**Solution:**

x + y = 12 ---------- (1)

2x + 3y = 32 ---------- (2)

From Equation (1), y = 12 - x

Substituting, y = 12 - x, in Equation (2),

2x + 3(12 - x ) = 32

=> 2x + 36 - 3x = 32

=> 2x - 3x = 32 - 36

=> -x = -4

=> x = 4

Substituting the value of x in y = 12 - x, we get

y = 12 - 4

= 8

Hence the solution is (x, y) = ( 4, 8).

The inequalities are those where the algebraic
expressions are connected by an inequality signs, less than (<),
greater than (>), less than equal to ($\leqslant$) , greater than equal to
($\geq$).

### Solved Example

**Question:**Find the value of x, 2x + 3

**<**7, for all natural numbers.

**Solution:**

= 2x + 3 - 3 < 7 - 3

= 2x < 4

= $\frac{2x}{2} < \frac{4}{2}$

= x < 2

=>

**$x = 1$**, is only the solution which satisfy the inequality.

=> 2 * 1 + 3 = 2 + 3 = 5 < 7.

Linear Equations are those
algebraic expression is equated to a particular constant or an
algebraic expression. In a linear equation there will be definite values
for the variables to satisfy the condition of the equation.

* For example,* Solution for the system of linear equations