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

Question 1:

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

First equation: x + y = 1

Case 1: Let us consider that x equals 0.

➝ x + y = 1

➝ 0 + y = 1

➝ y = 1

∴ First pair ⇒ (0, 1)

Case 2: Let us consider that x equals 1.

➝ x + y = 1

➝ 1 + y = 1

➝ y = 1 - 1

➝ y = 0

∴ Second pair ⇒ (1, 0)

Case 3: Let us consider that x equals -1.

➝ x + y = 1

➝ -1 + y = 1

➝ y = 1 + 1

➝ y = 2

∴ Third pair ⇒ (-1, 2)

Plot the points (0, 1), (1, 0), (-1, 2) on a graph, and join the points, the resulting line represents x + y = 1. [Blue line]

Second equation: 2x - 3y = 7

➝ 2x - 3y = 7

➝ 2x - 7 = 3y

➝ (2x - 7)/3 = y [We'll be substituing values in this equation]

Case 1: Let us consider that x equals 5.

➝ (2x - 7)/3 = y

➝ (2(5) - 7)/3 = y

➝ (10 - 7)/3 = y

➝ 3/3 = y

➝ y = 1

∴ First pair ⇒ (5, 1)

Case 2: Let us consider that x equals -1.

➝ (2x - 7)/3 = y

➝ (2(-1) - 7)/3 = y

➝ (-2 - 7)/3 = y

➝ -9/3 = y

➝ y = -3

∴ Second pair ⇒ (-1, -3)

Case 3: Let us consider that x equals -4.

➝ (2x - 7)/3 = y

➝ (2(-4) - 7)/3 = y

➝ (-8 - 7)/3 = y

➝ -15/3 = y

➝ y = -5

∴ Third pair ⇒ (-4, -5)

Plot the points (5, 1), (-1, -3), (-4, -5) on a graph, and join the points, the resulting line represents 2x - 3y = 7. [Red line]

Observations:

• From the graph, it's apparent that the two lines intersect at the point (2, -1), therefore the solution is (2, -1).

• The pair of lines are intersecting lines, they contain only one solution, and only have one point in common.

Question 2:

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

First equation: x + 2y = 4

Case 1: Let us consider that x equals 2.

➝ x + 2y = 4

➝ 2 + 2y = 4

➝ 2y = 4 - 2

➝ 2y = 2

➝ y = 1

∴ First pair ⇒ (2, 1)

Case 2: Let us consider that x equals 0.

➝ x + 2y = 4

➝ 0 + 2y = 4

➝ 2y = 4

➝ y = 4/2

➝ y = 2

∴ Second pair ⇒ (0, 2)

Case 3: Let us consider that x equals -2.

➝ x + 2y = 4

➝ -2 + 2y = 4

➝ 2y = 4 + 2

➝ y = 6/2

➝ y = 3

∴ Third pair ⇒ (-2, 3)

Plot the points (2, 1), (0, 2), (-2, 3) on a graph, and join the points, the resulting line represents x + 2y = 4. [Orange line]

Second equation: 2x + 4y = 12

➝ 2x + 4y = 12

➝ 2(x + 2y) = 12

➝ (x + 2y) = 12/2

➝ x + 2y = 6 [We'll be substituing values in this equation]

Case 1: Let us consider that x equals 0.

➝ x + 2y = 6

➝ 0 + 2y = 6

➝ 2y = 6

➝ y = 6/2

➝ y = 3

∴ First pair ⇒ (0, 3)

Case 2: Let us consider that x equals 2.

➝ x + 2y = 6

➝ 2 + 2y = 6

➝ 2y = 6 - 2

➝ y = 4/2

➝ y = 2

∴ Second pair ⇒ (2, 2)

Case 3: Let us consider that y equals 4.

➝ x + 2y = 6

➝ x + 2(4) = 6

➝ x + 8 = 6

➝ x = 6 - 8

➝ x = -2

∴ Third pair ⇒ (-2, 4)

Plot the points (0, 3), (2, 2), (-2, 4) on a graph, and join the points, the resulting line represents 2x + 4y = 12. [Black line]

Observations:

• From the graph, it's apparent that the two lines are parallel to each other, and do not intersect. Therefore, there is no solution.

• The pair of lines are parallel lines, therefore they do not contain any solutions, and do not hold any points in common.

Question 3:

(iii) 3x - y = 2 ; 9x - 3y = 6​

First equation: 3x - y = 2

Case 1: Let us consider that x equals 0.

➝ 3x - y = 2

➝ 3(0) - y = 2

➝ - y = 2

➝ y = -2

∴ First pair ⇒ (0, -2)

Case 2: Let us consider that x equals 1.

➝ 3x - y = 2

➝ 3(1) - y = 2

➝ 3 - y = 2

➝ -y = 2 - 3

➝ -y = -1

➝ y = 1

∴ Second pair ⇒ (1, 1)

Case 3: Let us consider that x equals 2.

➝ 3x - y = 2

➝ 3(2) - y = 2

➝ 6 - y = 2

➝ -y = 2 - 6

➝ -y = -4

➝ y = 4

∴ Third pair ⇒ (2, 4)

Plot the points (0, -2), (1, 1), (2, 4) on a graph, and join the points, the resulting line represents x + y = 1. [Blue line]

Second equation: 9x - 3y = 6​

➝ 9x - 3y = 6​

➝ 3(3x - y) = 6​

➝ 3x - y = 6​/3

➝ 3x - y = 2

➝ 3x - y = 2

This equation is the same as the first equation we've found coordinates for, so we can make use of the same three points for plotting purposes, which are (0, -2), (1, 1) and (2, 4). [Blue line]

Observations:

• From the graph, it's apparent that the two lines are co-incident lines, they have infinitely many solutions.

• The pair of lines are co-incident, and they contain only infinitely many solutions and have infinite points in common.

[Graphs attached, made with Desmos]

Answer 2

Question 1 :-

Given :-

$\begin{gathered} \sf \: x + y =1 \\ \sf 2x - 3y = 7 \\ \end{gathered}$

To find :- Solution of linear pair of equation graphically.

Solution :-

Step 1 :- Find points to plot x + y = 1

Put x = 0

0 + y = 1

y = 1

let it is A(0,1)

Put y = 0

x + 0 = 1

x = 1

let it is B(1,0)

Plot points on graph,make a straight line by joining these points.

Step 2 :- Find points to plot 2x - 3y = 7

Put x = 0

0 - 3y = 7

y = -2.3

let it is C(0,-2.3)

Put y = 0

2x + 0 = 7

x = 3.5

let it is D(3.5,0)

Plot points on graph,make a straight line by joining these points.

Graph is attatched.

The line intersects each other at (2,-1), this is the unique solution of these lines.

Final answer :-

(2,-1) is the solution of given equations.

Question 2 :-

Given :-

$\begin{gathered}\begin{gathered} \sf x +2y =4 \\ \sf 2x +4y = 12 \\ \end{gathered} \end{gathered}$

To find :- Solution of linear pair of equation graphically.

Solution :-

Step 1 :- Find points to plot x + 2y = 4

Put x = 0

0 + 2y = 4

y = 2

let it is point A(0,2)

Put y = 0

x + 0 = 4

x = 4

let it is point B(4,0)

Plot points on graph, make a straight line by joining these points.

Step 2 :- Find points to plot 2x + 4y = 12

Put x = 0

0 + 4 y = 12

y = 3

let it is point C(0,3)

Put y = 0

2x + 0 = 12

x = 6

let it is point D(6,0)

Plot points on graph, make a straight line by joining these points.

Graph is attatched.

It is clearly shown that both the lines are parallel,they never meet each other.

Therefore, these linear equation has no solution.

Final answer :-

Given pair of linear equations have no solution.

System of equations is inconsistent.

Question 3 :-

Given :-

$\begin{gathered}\begin{gathered}\begin{gathered}\sf 3x-y=2 \\ \sf 9x -3y = 6 \\ \end{gathered} \end{gathered}\end{gathered}$

To find :- Solution of linear pair of equation graphically.

Solution :-

Step 1 :- Find points to plot 3x - y = 2

Put x = 0

0 - y = 2

y = -2

let it is point A(0,-2)

Put y = 4

3x - 4 = 2

3x = 6

x = 2

let it is point B(2,4)

Plot points on graph, make a straight line by joining these points.

Step 2 :- Find points to plot 9x - 3y = 6

Put x = -1

-9 -3y = 6

y = -5

let it is point C(-1,-5)

Put y = 10

9x - 30 = 6

x = 4

let it is point D(4,10)

Plot points on graph, make a straight line by joining these points.

Graph is attatched.

One line is dashed and one is plane,so that it will distinguish.

It is clearly shown that both the lines are coincide (Overlapping) ; they meet at infinite time.

Therefore, these linear equation has infinite many solution.

Final answer :-

Given pair of linear equations have infinite many solution.

System of equations is consistent.