# Linear Equation In Two Variables Solution of TS & AP Board Class 9 Mathematics

###### Exercise 6.3

**Question 1.**

Draw the graph of each of the following linear equations.

2y = -x + 1

**Answer:**For graph, we’ll first make the table of solutions by putting some random values of x and thereafter we’ll find corresponding values of y and then we’ll plot these points on graph, join them and extend them in straight line to find the graph.

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

**Question 2.**Draw the graph of each of the following linear equations.

–x + y = 6

**Answer:**For graph, we’ll first make the table of solutions by putting some random values of x and thereafter we’ll find corresponding values of y and then we’ll plot these points on graph, join them and extend them in straight line to find the graph.

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

**Question 3.**Draw the graph of each of the following linear equations.

3x + 5y = 15

**Answer:**For graph, we’ll first make the table of solutions by putting some random values of x and thereafter we’ll find corresponding values of y and then we’ll plot these points on graph, join them and extend them in straight line to find the graph.

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

**Question 4.**

**Answer:**For graph, we’ll first make the table of solutions by putting some random values of x and thereafter we’ll find corresponding values of y and then we’ll plot these points on graph, join them and extend them in straight line to find the graph.

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

**Question 5.**Draw the graph of each of the following linear equations.

y = x

**Answer:**For graph, we’ll first make the table of solutions by putting some random values of x and thereafter we’ll find corresponding values of y and then we’ll plot these points on graph, join them and extend them in straight line to find the graph.

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

**Question 6.**Draw the graph of each of the following linear equations.

y = 2x

**Answer:**For graph, we’ll first make the table of solutions by putting some random values of x and thereafter we’ll find corresponding values of y and then we’ll plot these points on graph, join them and extend them in straight line to find the graph.

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

**Question 7.**Draw the graph of each of the following linear equations.

y = -2x

**Answer:**For graph, we’ll first make the table of solutions by putting some random values of x and thereafter we’ll find corresponding values of y and then we’ll plot these points on graph, join them and extend them in straight line to find the graph.

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

**Question 8.**Draw the graph of each of the following linear equations.

y = 3x

**Answer:**For graph, we’ll first make the table of solutions by putting some random values of x and thereafter we’ll find corresponding values of y and then we’ll plot these points on graph, join them and extend them in straight line to find the graph.

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

**Question 9.**Draw the graph of each of the following linear equations.

y = -3x

**Answer:**For graph, we’ll first make the table of solutions by putting some random values of x and thereafter we’ll find corresponding values of y and then we’ll plot these points on graph, join them and extend them in straight line to find the graph.

(Note: ∵ equation is linear graph will always be straight line.) Table of solutions for the given equation-

GRAPH:

**Question 10.**Answer the following question related to above graphs.

i) Are all these equations of the form y = mx, where m is a real number?

ii) Are all these graphs passing through the origin?

iii) What can you conclude about these graphs?

**Answer:**(i) Yes, all these are equations of the form y = mx, where m is a real number and m = 1,2,-2,3,-3 respectively in the above equations.

(ii) Yes, all these are graphs passing through the origin, i.e., pt. A in every graph

(iii) ∴ we can conclude that every graph of type y = mx passes through origin, where m is a real number.

**Question 11.**Draw the graph of the equation 2x + 3y = 11. Find from the graph value of y when x = 1

**Answer:**For graph, we’ll first make the table of solutions by putting some random values of x and thereafter we’ll find corresponding values of y and then we’ll plot these points on graph, join them and extend them in straight line to find the graph.

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

From the graph (pt. E) we can see that for x = 1, the y = 3.

(Note: Also we can put x = 1 in the given equation and can find the value of y-

We have,

At x = 1,

⇒ y = 3

**Question 12.**Draw the graph of the equation y - x = 2. Find from the graph

i) the value of y when x = 4

ii) the value of x when y = -3

**Answer:**For graph, we’ll first make the table of solutions by putting some random values of x and thereafter we’ll find corresponding values of y and then we’ll plot these points on graph, join them and extend them in straight line to find the graph.

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

i)the value of y when x = 4 is y = 6 (pt. E)

ii) the value of x when y = -3 is y = -5 (pt. F)

**Question 13.**Draw the graph of the equation 2x + 3y = 12. Find the solutions from the graph

i) Whose y-coordinate is 3

ii) Whose x-coordinate is -3

**Answer:**For graph, we’ll first make the table of solutions by putting some random values of x and thereafter we’ll find corresponding values of y and then we’ll plot these points on graph, join them and extend them in straight line to find the graph.

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

(i) From the graph, we can see that for y = 3 is pt. E and the

(ii) From the graph, we can see that for x = -3 is pt. F and the corresponding y = 6 for that.

**Question 14.**Draw the graph of each of the equations given below and also find the coordinates of the points where the graph cuts the coordinate axes

6x - 3y = 12

**Answer:**For graph, we’ll first make the table of solutions by putting some random values of x and thereafter we’ll find corresponding values of y and then we’ll plot these points on graph, join them and extend them in straight line to find the graph.

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

⇒ ∴ the pts. Where graph cuts the co-ordinate axis(i.e., where x = 0 and where y = 0) are pt. A = (2,0) and pt. B = (0,-4)

**Question 15.**Draw the graph of each of the equations given below and also find the coordinates of the points where the graph cuts the coordinate axes

- x + 4y = 8

**Answer:**For graph, we’ll first make the table of solutions by putting some random values of x and thereafter we’ll find corresponding values of y and then we’ll plot these points on graph, join them and extend them in straight line to find the graph.

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

⇒ ∴ the pts. Where graph cuts the co-ordinate axis(i.e., where x = 0 and where y = 0) are pt. A = (-8,0) and pt. B = (0,2).

**Question 16.**Draw the graph of each of the equations given below and also find the coordinates of the points where the graph cuts the coordinate axes

3x + 2y + 6 = 0

**Answer:**For graph, we’ll first make the table of solutions by putting some random values of x and thereafter we’ll find corresponding values of y and then we’ll plot these points on graph, join them and extend them in straight line to find the graph.

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

⇒ ∴ the pts. Where graph cuts the co-ordinate axis(i.e., where x = 0 and where y = 0) are pt. A = (-2,0) and pt. B = (0,-3).

**Question 17.**Rajiya and Preethi two students of Class IX together collected ₹ 1000 for the Prime Minister Relief Fund for victims of natural calamities. Write a linear equation and draw a graph to depict the statement.

**Answer:**Given that together Rajiya and Preethi collected Rs.1000.

Now, Let the amount collected by Rajiya be Rs. x and by Preethi be Rs. y.

∴ the linear equation will be-

⇒ x + y = 1000

For graph, we’ll first make the table of solutions by putting some random values of x and thereafter we’ll find corresponding values of y and then we’ll plot these points on graph, join them and extend them in straight line to find the graph.

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

**Question 1.**

Draw the graph of each of the following linear equations.

2y = -x + 1

**Answer:**

For graph, we’ll first make the table of solutions by putting some random values of x and thereafter we’ll find corresponding values of y and then we’ll plot these points on graph, join them and extend them in straight line to find the graph.

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

**Question 2.**

Draw the graph of each of the following linear equations.

–x + y = 6

**Answer:**

For graph, we’ll first make the table of solutions by putting some random values of x and thereafter we’ll find corresponding values of y and then we’ll plot these points on graph, join them and extend them in straight line to find the graph.

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

**Question 3.**

Draw the graph of each of the following linear equations.

3x + 5y = 15

**Answer:**

For graph, we’ll first make the table of solutions by putting some random values of x and thereafter we’ll find corresponding values of y and then we’ll plot these points on graph, join them and extend them in straight line to find the graph.

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

**Question 4.**

**Answer:**

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

**Question 5.**

Draw the graph of each of the following linear equations.

y = x

**Answer:**

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

**Question 6.**

Draw the graph of each of the following linear equations.

y = 2x

**Answer:**

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

**Question 7.**

Draw the graph of each of the following linear equations.

y = -2x

**Answer:**

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

**Question 8.**

Draw the graph of each of the following linear equations.

y = 3x

**Answer:**

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

**Question 9.**

Draw the graph of each of the following linear equations.

y = -3x

**Answer:**

(Note: ∵ equation is linear graph will always be straight line.) Table of solutions for the given equation-

GRAPH:

**Question 10.**

Answer the following question related to above graphs.

i) Are all these equations of the form y = mx, where m is a real number?

ii) Are all these graphs passing through the origin?

iii) What can you conclude about these graphs?

**Answer:**

(i) Yes, all these are equations of the form y = mx, where m is a real number and m = 1,2,-2,3,-3 respectively in the above equations.

(ii) Yes, all these are graphs passing through the origin, i.e., pt. A in every graph

(iii) ∴ we can conclude that every graph of type y = mx passes through origin, where m is a real number.

**Question 11.**

Draw the graph of the equation 2x + 3y = 11. Find from the graph value of y when x = 1

**Answer:**

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

From the graph (pt. E) we can see that for x = 1, the y = 3.

(Note: Also we can put x = 1 in the given equation and can find the value of y-

We have,

At x = 1,

⇒ y = 3

**Question 12.**

Draw the graph of the equation y - x = 2. Find from the graph

i) the value of y when x = 4

ii) the value of x when y = -3

**Answer:**

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

i)the value of y when x = 4 is y = 6 (pt. E)

ii) the value of x when y = -3 is y = -5 (pt. F)

**Question 13.**

Draw the graph of the equation 2x + 3y = 12. Find the solutions from the graph

i) Whose y-coordinate is 3

ii) Whose x-coordinate is -3

**Answer:**

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

(i) From the graph, we can see that for y = 3 is pt. E and the

(ii) From the graph, we can see that for x = -3 is pt. F and the corresponding y = 6 for that.

**Question 14.**

Draw the graph of each of the equations given below and also find the coordinates of the points where the graph cuts the coordinate axes

6x - 3y = 12

**Answer:**

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

⇒ ∴ the pts. Where graph cuts the co-ordinate axis(i.e., where x = 0 and where y = 0) are pt. A = (2,0) and pt. B = (0,-4)

**Question 15.**

Draw the graph of each of the equations given below and also find the coordinates of the points where the graph cuts the coordinate axes

- x + 4y = 8

**Answer:**

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

⇒ ∴ the pts. Where graph cuts the co-ordinate axis(i.e., where x = 0 and where y = 0) are pt. A = (-8,0) and pt. B = (0,2).

**Question 16.**

Draw the graph of each of the equations given below and also find the coordinates of the points where the graph cuts the coordinate axes

3x + 2y + 6 = 0

**Answer:**

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH:

⇒ ∴ the pts. Where graph cuts the co-ordinate axis(i.e., where x = 0 and where y = 0) are pt. A = (-2,0) and pt. B = (0,-3).

**Question 17.**

Rajiya and Preethi two students of Class IX together collected ₹ 1000 for the Prime Minister Relief Fund for victims of natural calamities. Write a linear equation and draw a graph to depict the statement.

**Answer:**

Given that together Rajiya and Preethi collected Rs.1000.

Now, Let the amount collected by Rajiya be Rs. x and by Preethi be Rs. y.

∴ the linear equation will be-

⇒ x + y = 1000

(Note: ∵ equation is linear graph will always be straight line.)

Table of solutions for the given equation-

GRAPH: