Linear Equations

The cost of 4 tables and 2 chairs cost Rs. 500 and the cost of 5 tables and 4 chairs cost Rs. 700, 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.

Different Forms

A linear equation does not involve any products or roots of variables. All variables have degree one. 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.

Linear Equations in one Variable

The only power of the variable is 1.  In a linear equation there will be definite values for the variables to satisfy the condition of the equation.

General form: ax + b = 0: where a $\neq$ 0

Linear Equations in Two Variables

Linear equations of two variables are of the form, ax + by + c = 0, (where a$\neq$ 0and b$\neq$0) also known as the general form of the straight line. 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.
Below are different forms of the linear equations:
  • Slope intercept form ; General form is  y = mx + c
  • Point–slope form; General form is y - y_1 = m(x - x_1)
  • Intercept form; General form is  x/x_0 + y/y_0 = 1

Where m = slope of a line; (x_0, y_0) intercept of x-axis and y-axis.

System of Linear Equations

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

This 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.

Graphing Linear Equations

  • 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:

Linear Equation Graph

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)


Graphing Linear Equations

Solving Linear Equations

How to solve linear Equations? Linear Equations of 2 or more variables can be solved by various methods. 
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.  To solve equations you have to be tricky and choose smartly any of the methods. Some of them are as follows:
  • Cross multiplication method
  • Method of substitution
  • Method of elimination
  • Matrix method
  • Determinant methods

Let us solve the set of equation using method of substitution:
x + y = 12 and 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

Linear equation, a statement involving one or more variables whose degree is one. 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 ($\geq$).
Examples:
Linear equations: 3x, 1/2x + 5 = 0, 2x + 1/2
Linear inequalities: 2x > 0, 2x + 8 < 6, 1/3x + 10 $\leq$ 5


Let us solve an inequality equation, Find the value of x, 2x + 7, for all natural numbers.
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.