Pair of Linear Equations in Two Variables Test

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Pair of Linear Equations in Two Variables Test - 13

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Question 1
1 / -0

Every linear equation in two variables has:

A
One solution

B
Two solutions

C
No solution

D
An infinite number of solutions

Solution

A linear equation in two variables is of the form, ax + by + c = 0.
Where geometrically it does represent a straight line and every point on this graph is a solution for a given linear equation.
As a line consists of an infinite number of points, A linear equation has an infinite number of solutions.
Hence, the correct option is (D).
Question 2
1 / -0

The pair of linear equations \(4x – 6y = 9\) and \(2x – 3y = 8\) are:

A
Inconsistent

B
Consistent

C
Dependent

D
None of these

Solution

Given:
\(a_{1}=4, a_{2}=2, b_{1}=-6, b_{2}=-3, c_{1}=9\) and \(c_{2}=8\)
Here \(\frac{a_{1}}{a_{2}}=\frac{4}{2}=\frac{2}{1}, \frac{b_{1}}{b_{2}}=\frac{-6}{-3}=\frac{2}{1}, \frac{c_{1}}{c_{2}}=\frac{9}{8}\)
\(\because \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}\)
Therefore, the pair of given linear equations are inconsistent.
Hence, the correct option is (A).
Question 3
1 / -0

A system of two linear equations in two variables is inconsistent, if their graphs

A
do not intersect at any point

B
coincide

C
cut the x – axis

D
intersect only at a point

Solution

If a consistent system has an infinite number of solutions, it is dependent . When you graph the equations, both equations represent the same line. If a system has no solution, it is said to be inconsistent . The graphs of the lines do not intersect, so the graphs are parallel and there is no solution.
Question 4
1 / -0

The pair of linear equations 3x + 2y = 5 and 2x – 3y = 7 are

A
inconsistent

B
dependent(consistent)

C
consistent

D
none of these

Solution

Given:
\(a_{1}=3, a_{2}=2, b_{1}=2, b_{2}=-3, c_{1}=5\) and \(c_{2}=7\)
Here \(\frac{a_{1}}{a_{2}}=\frac{3}{2}, \frac{b_{1}}{b_{2}}=\frac{2}{-3}\)
\(\because \frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}\)
Therefore, the pair of given linear equations are consistent.
Question 5
1 / -0

A system of two linear equations in two variables has infinitely many solutions, if their graphs

A
do not intersect at any point

B
intersect only at a point

C
cut the x –axis

D
coincide with each other.

Solution

A system of two linear equations in two variables has infinitely many solutions, if their graphs coincide with each other and all the points in one line are common with the other line.
Question 6
1 / -0

Ten students of class X took part in Mathematics quiz. The number of girls is 4 more than that of the boys. The algebraic representation of the above situation is

A
y = x + 4 and x = 10 - y

B
x = y - 12 and y = 6 + x

C
x – y = 10 and x + y = 4

D
none of these

Solution

Let number of boys = x
Number of girls = y
Given that total number of student is 10 so that
x + y = 10
subtract y on both side we get,
x = 10 – y
Given,
the number of girls is 4 more than the number of boys
So that,
y = x + 4
Question 7
1 / -0

The lines representing the pair of equations 9x + 3y + 12 = 0 and 18x + 6y + 24 = 0

A
are coincident

B
intersect at a point

C
intersected at two points

D
are parallel

Solution

Given:
\(a_{1}=9, a_{2}=18, b_{1}=3, b_{2}=6, c_{1}=12\) and \(c_{2}=24\)
Here \(\frac{a_{1}}{a_{2}}=\frac{9}{18}=\frac{1}{2}, \frac{b_{1}}{b_{2}}=\frac{3}{6}=\frac{1}{2}, \frac{c_{1}}{c_{2}}=\frac{12}{24}=\frac{1}{2}\)
\(\because \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)
Therefore, the lines are coincident.
Question 8
1 / -0

The lines representing the pair of equations 6x – 3y + 10 = 0 and 2x – y + 9 = 0

A
Intersected at two points

B
are parallel

C
are coincident

D
intersect at a point

Solution

Given:
\(a_{1}=6, a_{2}=2, b_{1}=-3, b_{2}=-1, c_{1}=10\) and \(c_{2}=9\)
Here \(\frac{a_{1}}{a_{2}}=\frac{6}{2}=\frac{3}{1}, \frac{b_{1}}{b_{2}}=\frac{-3}{-1}=\frac{3}{1}, \frac{c_{1}}{c_{2}}=\frac{10}{9}\)
but \(c_ 1 / c _2=10 / 9\)
\(\because \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}\)
Therefore, the lines are parallel.
Question 9
1 / -0

A system of two linear equations in two variables is dependent consistent if their graphs ______________.

A
Intersect only at a point

B
Coincide each other

C
Do not intersect at any point

D
None of the above

Solution

A system of two linear equations in two variables is dependent consistent if their graphs do not intersect at any point.
If a consistent system has an infinite number of solutions, it is dependent. When we graph the equations, both equations represent the same line. If a system has no solution, it is said to be inconsistent. The graphs of the lines do not intersect, so the graphs are parallel and there is no solution.
Hence, the correct option is (C).
Question 10
1 / -0

The pair of linear equations 5x – 3y = 11 and – 10x + 6y = – 22 are

A
consistent

B
none of these

C
inconsistent

D
dependent(consistent)

Solution

Given:
\(a_{1}=5, a_{2}=-10, b_{1}=-3, b_{2}=6, c_{1}=11\) and \(c_{2}=-22\)
Here \(\frac{a_{1}}{a_{2}}=\frac{5}{-10}=-\frac{1}{2}, \frac{b_{1}}{b_{2}}=\frac{-3}{6}=-\frac{1}{2}, \frac{c_{1}}{c_{2}}=\frac{11}{-22}=-\frac{1}{2}\)
\(\because \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)
Therefore, the pair of given linear equations are dependent consistent.
Question 11
1 / -0

The lines representing the pair of equations x + 3y = 6 and 2x – 3y = 12 intersect at

A
(1, 6)

B
(6, 1)

C
(6, 0)

D
(0, 6)

Solution

Here are the two solutions of each of the given equations.

In the graph, both given lines are intersect at the point \(B(6,0)\)
Question 12
1 / -0

A system of two linear equations in two variables has no solution, if their graphs

A
do not intersect at any point

B
intersect only at a point

C
coincide

D
cut the x – axis

Solution

A system of two linear equations in two variables is inconsistent, if their graphs do not intersect at any point.
In this case, a pair of lines represented by the system are parallel to each other. so they do not intersect each other at any point.
then there are no solutions that are true for both equations.
Question 13
1 / -0

A system of two linear equations in two variables is consistent, if their graphs

A
cut the x –axis

B
intersect only at a point or they coincide with each other

C
do not intersect at any point

D
coincide

Solution

A system of two linear equations in two variables is consistent, if their graphs intersect only at a point, because it has a unique solution or they may coincide with each other giving infinite solutions.
Question 14
1 / -0

The lines representing the pair of equations 5x – 4y + 8 = 0 and 7x + 6y – 9 = 0

A
intersect at a point

B
are parallel

C
are coincident

D
none of these

Solution

Two lines
\(a_{1} x+b_{1} y+c_{1}=0\)
\(a_{2} x+b_{2} y+c_{2}=0\)
parallel if \(a_{1} / a_{2}=b_{1} / b_{2}\)
Coincident if \(a_{1}/ a _{2}= b _{1} / b _{2}= c _{1} / c _{2}\)
Intersect ot a point if \(a_{1} / a_{2} \neq b_{1} / b_{2}\)
Lets check for
\(5 x-4 y+8=0\)
\(7 x+6 y-9=0\)
\(a _{1} / a _{2}=5 / 7\)
\(b _{1} / b _{2}=-4 / 6\)
\(c _{1} / c _{2}=8 /-9\)
\(5 / 7 \quad \neq-4 / 6\)
\(a_{1} / a_{2} \neq b_{1} / b_{2}\)
\(\Rightarrow\) lines intersect at a point
Question 15
1 / -0

A system of two linear equations in two variables has a unique solution, if their graphs

A
cut the x – axis

B
do not intersect at any point

C
intersect only at a point

D
coincide

Solution

Number of solutions of a system of two linear equations in two variables are equal to number of common points between the graphs of given linear equations.

If a system has unique solution then their graphs must intersect in only one point.

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Pair of Linear Equations in Two Variables Test - 13

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Pair of Linear Equations in Two Variables Test - 13

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SHARING IS CARING

Question 1

One solution

Two solutions

No solution

An infinite number of solutions

Solution

Question 2

Inconsistent

Consistent

Dependent

None of these

Solution

Question 3

do not intersect at any point

coincide

cut the x – axis

intersect only at a point

Solution

Question 4

inconsistent

dependent(consistent)

consistent

none of these

Solution

Question 5

do not intersect at any point

intersect only at a point

cut the x –axis

coincide with each other.

Solution

Question 6

y = x + 4 and x = 10 - y

x = y - 12 and y = 6 + x

x – y = 10 and x + y = 4

none of these

Solution

Question 7

are coincident

intersect at a point

intersected at two points

are parallel

Solution

Question 8

Intersected at two points

are parallel

are coincident

intersect at a point

Solution

Question 9

Intersect only at a point

Coincide each other

Do not intersect at any point

None of the above

Solution

Question 10

consistent

none of these

inconsistent

dependent(consistent)

Solution

Question 11

(1, 6)

(6, 1)

(6, 0)

(0, 6)

Solution

Question 12

do not intersect at any point

intersect only at a point

coincide

cut the x – axis

Solution