Question

Solve the following pair of linear equations graphically. Also write the observations.

(i) x + y = 1 ; 2x - 3y = 7

(ii) x + 2y = 4 ; 2x + 4y = 12

(iii) 3x - y = 2 ; 9x - 3y = 6​

Answer 1

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Solve the following pair of linear equations graphically. Also write the observations.

(i) x + y = 1 ; 2x - 3y = 7

(ii) x + 2y = 4 ; 2x + 4y = 12

(iii) 3x - y = 2 ; 9x - 3y = 6

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i) 2x=7-3y

X=7-3y/2

Substituting the value of x in x+y=1

7-3y/2+y=1

7-3y+2y=2 Putting the value of y in 2x-3y=7

7-(-1y)=2

1y=2-7

Y=-5

2x-(-5)=7

2x+5=7

2x=7-5

X=1

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ii) The two linear equation in two variable are 

1.x- 2 y=4

2. 2 x - 4 y=12

So, if two lines are parallel then their slopes are equal

And , if they are coincident , Line 1= K × Line 2 , or Line 2= P × Line 1, where P and K are any real number.

But if you will observe the two lines 

2×Line 1= 2 x - 4 y-8≠Line 1=2 x - 4 y-12

So, these two lines are not coincident.

Now, Line 1 can be written as, 2 y= x -4

Line , 2 can be written as

4 y=2 x -12

Comparing with slope intercept form of line, y= m x + c

Slope of line 1 

Slope of line 2 

→As, slopes are equal. So, these two lines are Parallel.Also,

→Coefficient of x of line 2 = 2 × Coefficient of x of line 1

→Coefficient of y of line 2=2× Coefficient of y of line 2

_________________

iii) Given : System of equations 3x-y=2 , 9x-3y= 63x−y=2,9x−3y=6

To find : Show graphically that the system of equations has infinitely solutions.

Solution :

We have given the system of equations

3x-y=23x−y=2  ..(1)

9x-3y= 69x−3y=6  ..(2)

If we reduced the equation (2) by dividing 3 both side,

3x-y= 23x−y=2  ..(2)

The equation (2) became equation (1),

i.e. Both equations are equal so they have infinitely many solutions.

We plot both the equations represented by a single line.

Graphically we say that both equation forming a single line means they have infinitely many solutions.

________________

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Answer 2

Step-by-step explanation:

Solutions :-

(I)

Given pair of linear equations in two variables are

x + y = 1

=>x+y-1 = 0----(1)

On Comparing this with a1x+b1y+c1 = 0,

a1 = 1 , b1 = 1 , c1 = -1

2x - 3y = 7

=>2x-3y-7 = 0 -----(2)

On Comparing this with a2x+b2y+c2= 0,

a2 = 2 ,b2 = -3 ,c2 = -7

a1/a2 = 1/2

b1/b2 = 1/-3 = -1/3

c1/c2 = -1/-7 = 1/7

a1/a2 ≠ b1/b2 ≠ c1/c2

So given pair of linear equations in two variables are Consistent and dependent lines or Intersecting lines with an unique solution.

-» The graphs of two equations are straight lines and they are Intersecting at (2,-1)

-» Scale : On X-axis 1 cm = 1 unit

On y-axis 1 cm = 1 unit.

-» The solution = (2,-1)

ii)

Given pair of linear equations in two variables are

x + 2y = 4 ; 2x + 4y = 12

=>x+2y-4 = 0---(1)

On Comparing this with a1x+b1y+c1 = 0,

a1 = 1 , b1 = 2 , c1 = -4

2x + 4y = 12

=>2x+4y-12 = 0 -----(2)

On Comparing this with a2x+b2y+c2= 0,

a2 = 2 ,b2 = 4 ,c2 = -12

a1/a2 = 1/2

b1/b2 = 2/4 = -1/2

c1/c2 = -4/-12=1/3

a1/a2 = b1/b2 ≠ c1/c2

So given pair of linear equations in two variables are Inconsistent lines or parallel lines with no solution.

-» The graphs of two equations are straight lines and they are not Intersecting.

-»They are parallel lines

-» Scale : On X-axis 1 cm = 1 unit

On y-axis 1 cm = 1 unit.

-» They have no solution

iii)

Given pair of linear equations in two variables are

3x - y = 2 ; 9x - 3y = 6

3x-y = 2

=> 3x-y-2 = 0---(1)

On Comparing this with a1x+b1y+c1 = 0,

a1 = 3, b1 = -1 , c1 = -2

9x-3y=6

=>9x-3y-6= 0 -----(2)

On Comparing this with a2x+b2y+c2= 0,

a2 = 9 ,b2 = -3 ,c2 = -6

a1/a2 = 3/9=1/3

b1/b2 = -1/-3= 1/3

c1/c2 = -2/-6 = 1/3

a1/a2 = b1/b2 = c1/c2

So given pair of linear equations in two variables are consistent lines and dependent lines or Coincident lines with so many solutions.

-» The graphs of two equations are straight lines and they are coincident.

-»They are consistent lines

-» Scale : On X-axis 1 cm = 1 unit

On y-axis 1 cm = 1 unit.

-» They have infinitely number of many solutions.

Used formulae:-

a1x+b1y+c1 =0 and a2x+b2y+c2 = 0 are linear equations in two variables,

-» If a1/a2 ≠b1/b2 then they are Intersecting lines or Consistent and Independent lines with an unique solution.

-»If a1/a2 =b1/b2 ≠c1/c2 then they are parallel lines lines or inConsistent lines with no solution.

-»If a1/a2 =b1/b2 =c1/c2 then they are consistent and dependentl lines or coincident lineswith infinitely number of many solutions.