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Linear Equations

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.

Linear Equations Solver

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A linear equation does not involve any products or roots of variables. All variables occur only to the first power and do not appear. There are many ways of writing linear equations, but they usually have constants and variables.

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 ax + 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

Linear equations in two variables

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Linear equations of two variables are of the form, ax + by + c = 0. To solve for the variables x and y, we need to have two equations. Otherwise for every values of x there will be a corresponding value of y.  Hence a single equation has infinite number of solutions and each solution is the point on the line.

Solved Example

Question: Solve x + y = 5
Solution:
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)


  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.

 

Graphing Linear Equations

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  • 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.
Let us draw the graph for the equation, x + y = 5

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




System of Linear Equations

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System of linear equations are of the form:

a1x  + b1 y + c1 = 0  and

a2 x  + b2 y + c2 = 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.

Solving Linear Equations

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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:
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).
 

Linear Equations and Inequalities

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Linear Inequalities:

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 < 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.
 

Linear equations in one variable

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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

x + y = 12, 2x + 3y = 32 is x = (4, 8)