**NCERT Class 10 Maths Lab Manual – Linear Equations**

**Objective **

To verify the conditions for consistency of a system of linear equations in two variables by graphical representation.

**Linear Equation**

An equation of the form ax + by + c = 0, where a, b, c are real numbers, a ≠ 0, b ≠ 0 and x, y are variables; is called a linear equation in two variables.

You can also download **NCERT Solutions For Class 10 Maths** to help you to revise complete syllabus and score more marks in your examinations.

**Prerequisite Knowledge**

- Plotting of points on a graph paper.
- Condition of consistency of lines parallel, intersecting, coincident,

**Materials Required**

Graph papers, fevicol, geometry box, cardboard.

**Procedure**

Consider the three pairs of linear equations

**1stpair:** 2x-5y+4=0, 2x+y-8 = 0

**2nd pair:** 4x + 6y = 24, 2x + 3y =6

**3rd pair:** x-2y=5, 3x-6y=15

- Take the 1st pair of linear equations in two variables, e.g., 2x – 5y +4=0, 2x +y-8 = 0.
- Obtain a table of at least three such pairs (x, y) which satisfy the given equations.

- Plot the points of two equations on the graph paper as shown in fig. (i).

- Observe whether the lines are intersecting, parallel or coincident. Write the values in observation table.

Also, check ;\(\frac { { a }_{ 1 } }{ { a }_{ 2 } } ;\frac { { b }_{ 1 } }{ { b }_{ 2 } } ;\frac { { c }_{ 1 } }{ { c }_{ 2 } }\) - Take the second pair of linear equations in two variables

- Repeat the steps 3 and 4.

- Take the third pair of linear equations in two variables,i.e. x-2y=5, 3x-6y=15

- Repeat steps 3 and 4

Obtain the condition for two lines to be intersecting, parallel or coincident from the observation table by

comparing the values of \(\frac { { a }_{ 1 } }{ { a }_{ 2 } } ,\frac { { b }_{ 1 } }{ { b }_{ 2 } } and\frac { { c }_{ 1 } }{ { c }_{ 2 } }\)

**Observation**

Students will observe that

- for intersecting lines, \(\frac { { a }_{ 1 } }{ { a }_{ 2 } } \neq \frac { { b }_{ 1 } }{ { b }_{ 2 } } \)
- for parallel lines, \(\frac { { a }_{ 1 } }{ { a }_{ 2 } } =\frac { { b }_{ 1 } }{ { b }_{ 2 } } \neq \frac { { c }_{ 1 } }{ { c }_{ 2 } }\)
- for coincident lines, \(\frac { { a }_{ 1 } }{ { a }_{ 2 } } =\frac { { b }_{ 1 } }{ { b }_{ 2 } } =\frac { { c }_{ 1 } }{ { c }_{ 2 } }\)

**Result**

The conditions for consistency of a system of linear equations in two variables is verified.

**Learning Outcome**

Students will learn that some pairs of linear equations in two variables have a unique solution (intersecting lines), some have infinitely many solutions (coincident lines) and some have no solutions (parallel lines).

**Activity Time**

Perform the same activity by drawing graphs of x-y+1=0 and 3x + 2y – 12 =0. Show that there is a unique solution. Also from the graph, calculate the area bounded by these linear equations and x-axis.

**Viva Voce**

**Question 1.**

What is the equation of a line parallel to x-axis ?

**Answer:**

y = a, where a is any constant.

**Question 2.**

What is the equation of a line parallel to y-axis ?

**Answer:**

x = b, where b is any constant.

**Question 3.**

If x = 0 and y = 0, where would the point lie on graph ?

**Answer:**

At origin (0,0)

**Question 4.**

What is the condition for inconsistent and consistent solution for the system of linear equations ?

**Answer:**

Linear equations are

\({ a }_{ 1 }x+{ b }_{ 1 }y+{ c }_{ 1 }=0\\ { a }_{ 2 }x+{ b }_{ 2 }y+{ c }_{ 2 }=0\)

Inconsistent solution, \(\frac { { a }_{ 1 } }{ { a }_{ 2 } } =\frac { { b }_{ 1 } }{ { b }_{ 2 } } \neq \frac { { c }_{ 1 } }{ { c }_{ 2 } }\)

Consistent solution,

(i) \(\frac { { a }_{ 1 } }{ { a }_{ 2 } } \neq \frac { { b }_{ 1 } }{ { b }_{ 2 } } \)

(ii) \(\frac { { a }_{ 1 } }{ { a }_{ 2 } } =\frac { { b }_{ 1 } }{ { b }_{ 2 } } =\frac { { c }_{ 1 } }{ { c }_{ 2 } }\)

**Question 5.**

Is the pair of linear equations 2x + 3y – 9 = 0 and 4x + 6y -18 = 0, consistent ?

**Answer:**

Here \(\frac { { a }_{ 1 } }{ { a }_{ 2 } } =\frac { { b }_{ 1 } }{ { b }_{ 2 } } =\frac { { c }_{ 1 } }{ { c }_{ 2 } } =\frac { 1 }{ 2 }\)

⇒ Given system of equations is consistent and has infinitely many solutions.

**Question 6.**

For what value of p does the pair of linear equations given below has unique solution ?

4x + 8 = 0, 2x + 2y + 2 = 0

**Answer:**

For unique solution, \(\frac { { a }_{ 1 } }{ { a }_{ 2 } } \neq \frac { { b }_{ 1 } }{ { b }_{ 2 } } \Longrightarrow \frac { 4 }{ 2 } \neq \frac { p }{ 2 } \Longrightarrow { p\neq { 4 } }\)

**Question 7.**

What does the graph of a linear equation represent ?

**Answer:**

A straight line

**Question 8.**

If the graphical solutions of two linear equations of two lines are parallel to each other in plane, then what type of solution do they have ?

**Answer:**

No solution

**Question 9.**

If the graphical solutions of two linear equations of two lines intersect in a plane, then what type of the solution do they have ?

**Answer:**

Unique solution

**Multiple Choice Questions**

**Question 1.**

Is x= -1, y=5 a solution of the equation 4x + 3y = 11 ?

(a) yes

(b) no

(c) can’t say

(d) none of these

**Question 2.**

Equations 5x + 2y=16 and 7x-4y = 2 have:

(a) no solution

(b) a unique solution

(c) infinitely many solutions

(d) none of these

**Question 3.**

Equations —3x + 4y = 5 and \(\frac { 9 }{ 2 } x-6y=\frac { 15 }{ 2 }\)

(a) a unique solution

(b) infinitely many solutions

(c) no solution

(d) none of these

**Question 4.**

Equations -3x + 4y = 5 and \(\frac { 9 }{ 2 } x-6y+\frac { 15 }{ 2 }=0\) have:

(a) many solutions

(b) a unique solution

(c) no solution

(d) none of these

**Question 5.**

Condition for the system of linear equations ax + by = c; lx + my = n to have a unique solution is:

(a) am ≠ bl

(b) am = bl

(c) \(\frac { a }{ l } =\frac { b }{ m } =\frac { c }{ n }\)

(d) none of these

**Question 6.**

When \({ l }_{ 1 }\) and \({ l }_{ 2 }\) are parallel lines, then the graphical solution of system of linear equations has

(a) many solutions

(b) no solution

(c) a unique solution

(d) none of these

**Question 7.**

When lines \({ l }_{ 1 }\) and \({ l }_{ 2 }\) are coincident, then the graphical solution of system of linear equations has

(a) infinitely many solutions

(b) a unique solution

(c) no solution

(d) parallel lines

**Question 8.**

Values of x and y for the pair of linear equations x+y=14 and x-y = 4 are respectively

(a) 9 and 5

(b) 5 and 9

(c) 5 and 5

(d) 9 and 9

**Question 9.**

The difference between two numbers is 26 and one number is three times the other. The numbers are

(a) 39 and 12

(b) 39 and 13

(c) 38 and 13

(d) 13 and 13

**Question 10.**

In a cyclic quadrilateral BACD, ∠A = (2x – 4)°, ∠B = (y + 5)°, ∠C = (2y + 10)° and ∠D = (4x – 2)°. Find four angles.

(a) ∠A = 58°, ∠B = 60°, ∠C = 120°, ∠D = 122°

(b) ∠A = 65°, ∠B = 55°, ∠C = 115°, ∠D = 125°

(c) ∠A = 70°, ∠B = 110°, ∠C = 55°, ∠D = 125°

(d) ∠A = 65°, ∠B = 55°, ∠C = 110°, ∠D = 127°

**Answers**

- (a)
- (b)
- (c)
- (a)
- (a)
- (b)
- (a)
- (a)
- (b)
- (a)

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