# Writing Linear Equations

## 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:

#### 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 = $\frac{c}{b}$

#### 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.

## 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.

#### 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

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