The cost of 4 tables and 2 chairs cost 500 dollars and the cost of 5 tables and 4 chairs cost 700 dollars, what will be the costs of 10 chairs and 10 tables. When we come across situations like this we need to frame the corresponding equations. Here we assume a variable for the cost of a table and the cost of a chair. We then solve the equations for the variables.

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.

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.

- One variable is

- Two variables is a

- Three variables is

- Cross multiplication method
- Method of substitution
- Method of elimination
- Matrix method
- Determinant methods

Solved Example

We have x + y = 5

=> 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)

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.

=> 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)

a

a

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.

Let us discuss the method of substitution:

Solved Example

x + y = 12,

2x + 3y =32

Given

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).

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

2x + 3 < 7

= 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.

= 2x + 3 - 3 < 7 - 3

= 2x < 4

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

= x < 2

=>

=> 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