Introduction:

              We have already discussed the steps for solving linear equations of one variable and two variables. Let us discuss the various methods of representing a linear equation in two variables. 

Writing linear equations in standard form:

The standard form of linear equation of two variables, is ax + by = c, where a, b, c are real numbers and a and b both not equal to 0.
We are very much aware that the graph of the linear equation of two variables is a straight line on the two dimensional graph.
The standard form is very much useful to find the x and y intercepts of the graph of the equation.
Let us discuss an example:

Write the standard form of the linear equation, 2x + 3y = 10 = 0. Hence find the x and y intercepts.

Solution:

                              We have 2x + 3y -12 = 12
To write this in the standard form, ax + by = c, let us eliminate the constant term -10 from the left hand side.

                                   2x + 3y - 12 + 12 = 0 + 12
                               =>        2x + 3y       = 12

To find the x-intercept : The x-intercept is the point where the line meets the x-axis. (i.e) on the x-axis, y = 0

Therefore substituting y = 0 in the above standard form, we get
                                             2x + 3 (0) = 12
                                          => 2x  + 0   = 12
                                          =>          2x = 12
                                          =>            x = 12/2 = 6
Therefore the co-ordinate of x-intercept is ( 6,0)

To find the y-intercept : The y-intercept is the point where the line meets the y-axis. (i.e) on the the y-axis, x = 0.

Therefore substituting x = 0, in the above standard form,
                                             2 (0) + 3y = 12
                                         =>     0 + 3y = 12
                                         =>           3y = 12
                                         =>             y = 12/3
                                                             = 4
Therefore the co-ordinate of y-intercept is ( 0,4)

Writing linear equations in slope intercept form:


The slope intercept form of the linear equation of two variables is,
                            y = mx + c, where 'm' is the slope and 'c' the y-intercept of the line in the graph.

Writing slope-intercept form from the standard form:
 We have the standard form as ax + by = c

                                                     by = -ax + c
                                            
                                      $\frac{by}{b}$ = - $\frac{ax}{b}$  +  $\frac{c}{b}$ [ dividing both each term by a on both sides ]
                                                       y = - $\frac{a}{b}$ x + $\frac{c}{b} $, which is the slope-intercept form.
Comparing this with the slope intercept form,
                                                       y = mx + c,
                           we get the slope   m = - $ \frac{a}{b}$ and
                            the y-intercept is  c = $\frac{c}{b}$

Example: Express the equation 3x + 5y = 15 in slope intercept form. Also find the x and y-intercepts of the equation.

Solution:

              We have 3x + 5y = 15
                                   5y = - 3x + 15
                                     y = - $\frac{3}{5}$ + $\frac{15}{5}$
                                     y = -  $\frac{3}{5}$ + 3 , which is of the form y = mx + c, the slope - intercept form.

Comparing we get, the slope m = $\frac{3}{5}$ , and  the y-intercept c = 3

To find the x- intercept:

We have 3x + 5y = 15
Substituting y = 0, we get 3x + 5 ( 0) = 15
                                      => 3x  + 0 = 15
                                      =>        3x = 15
                                      =>          x = 15/3
                                                       = 5
The co-ordinate of the x-intercept is ( 5,0)

To find the y-intercept :

            Substituting x = 0 in 3x + 5y = 15, we get,
                                      3 (0 ) + 5y = 15
                              =>          0 + 5y = 15
                              =>                5y = 15
                              =>                  y = 15/5
                                                       = 3
The co-ordinate of the y-intercept is ( 0, 3)


Writing linear equations from graphs:

To write the linear equations from the graph, we should remember the intercept form of the linear equation.
If  a and b are the x and y - intercepts of a graph of a line, then the equation of a line is given by,
                                                                 $\frac{x}{a}$ + $ \frac{y}{b} $ = 1
For example, if a line meets the x-axis at x = 4 and y-axis at y = -3, then the equation of the line is $\frac{x}{4}$  + $ \frac{y}{-3}$ = 1

Hence the above equation can be written as, $\frac{x}{4}$ - $ \frac{y}{3}$ = 1
Multiplying both sides by the LCM of the denominations = 12,
                                                         12 ( $\frac{x}{4}$) - 12 $(\frac{y}{3})$ = 12(1)
                                                             =>     3x  - 4y = 12, which is the standard form of the equation.

graph2

Writing linear equations word problems: 

While  solving linear equations using word problems we  should remember the following points.
Step 1: Read the problem carefully and identify the unknown quantities. Assume these quantities a variable name x, y.
Step 2: Read the problem carefully and list the given quantities / table the given quantities.
Step 3: Read the problem again and form the two equations.
Step 4: Solve the two simultaneous equations for the variables x and y.
Note: Elimination method or cross multiplication method is the most suitable method for solving for the variables.

Example : Ten years ago, father was twelve times as old as his son and ten years hence, he will be twice as old as his son will be. Find their present ages.

Solution:

           Let us assume the present age of father as x years and that of the son's y years.
Let us table the given data.

  Father's age 
  Son's Age   Relation (or) Equation
   2 yrs ago     x - 10      y - 10
 Father's age = 12 (Son's age)
            x - 10 = 12( y - 10 )
  Present age 
      x
       y
 
 10 yrs hence    x + 10   
   y + 10
 Father's age = 2(Son's age)
         x + 10 = 2 ( y + 10)

Hence from the above table, we have the two equations,
                                    x - 10 = 12 ( y - 10) ------------------------(1)
                                   x + 10 = 2 ( y + 10) --------------------(2)
Let us simplify and write the equations (1) and (2) in standard form,
                                    x - 10 = 12 y - 120
                          => x - 12y    = -120 + 10 [ by transposition ]
                          =>  x - 12y   = -110 ------------------------(1)
                                   x + 10 = 2 ( y + 10 )
                                   x + 10 = 2y + 20
                                   x - 2y  = 20 - 10
                                  x - 2y   = 10 -----------------------(2)
                                  x - 12 y = - 110
                              ______________________
 Subtracting (1) from (2), we get                                                                        
                                       10 y =  120
                        =>               y = 120/10
                                              = 12
Substituting y = 12 in Equation (1), we get,
                                                                               x - 12 y = -110
                                                                    =>  x - 12 ( 12) = -110
                                                                    =>     x - 144    = -110
                                                                    =>                x = 144 - 110
                                                                                           = 34
                                                                   =>                 x = 34
Therefore, Father's Present age = 34 yrs and Son's Present age = 12 yrs

Practice Questions:

1. Express the equation , 3y + 8x -  24 = 0 in standard form. Hence find the x and y intercepts. Also express the equation in intercept form.

2. Points A and B are 90 km apart from each other on a highway. A car starts from A and another from B at the same time. If they go in the same direction they meet in 9 hrs and if they go in the opposite direction they  meet in 9/7 hrs. Find their speeds.

3. Study the given graph and write the equation in
a. Intercept form.
b. slope-intercept form
c. standard form
graph3

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