Algebra – Welcome

Linear equation in one variable

1) 3x + 6 = 150, what is the value of 15x + 30

OPTIONS: A) 720 B) 749 C) 750 D) 705

Solution:

3x + 6 = 150

3x = 150 – 6

3x = 144

x = 144 / 3 = 48

The value of 15x + 30 = 15 * 48 + 30 = 720 + 30 = 750

ANS: C

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2) If 3y – 4 = 5, what is the value of y + 6?

OPTIONS: A) 9 B) 6 C) 3 D) -9

ANS: A

Solution

3y – 4 = 5 given

Rearrange: 3y = 5 + 4 = 9

3y = 9

Divide by 3 on both sides:

3y / 3 = 9 / 3; y = 3

Substitute y = 3 in y + 6:

y + 6 = 3 + 6 = 9

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3) If 2x + 6 = 3x – 4, find the value of x.

OPTIONS: A) -10 B) 10 C) 6 D) 5

ANS: B

Solution

2x + 6 = 3x – 4

Rearrange: 2x – 3x = -4 – 6

-x = -10

x = 10

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4) If X + 2 = 11, what is the value of X²

OPTIONS: A) 91 B) 81 C) 71 D) 101

ANS: B

Solution

X + 2 = 11

Rearrange: X = 11 – 2 = 9

Squaring on both sides:

X² = 9²

X²

= 81

5) If a – x =- 4 ,What is the value of a – x +10.?

a – x = -4 (Given)

Add 10 0n both side

a – x + 10 = -4 +10 = 6

So a – x +10 = 6

ANS A) 6 B) – 6 C ) 7 D ) 8

Corrcet ANS A

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6) If 4 (x – 1) = 2o, What is the Value of X ?

4( x – 1 ) = 20 (given)

Open the parenthesis

4x – 4 = 20

Rearrange 4x = 20 + 4 =24 Divide by 4 on both side

4x/4 = 20/4

X = 5

ANS A) 5 B) 6 C) 9 D) 3

Correct ANS A

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7) if a = x + 5 and b = x – 6 , What is the value of a – b ?

Substitue the value of a and b in a – b

Then x+ 5 – ( x – 6 )

Simplify x + 5 -x + 6 ( open the parenthesis)

= 11

So a – b = 11

ANS A) 11 B) 12 c) 15 D) -10

Correct ANS A

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8) the sum of three consecutive even integer is 12.

What are three even integers?

General form of three consecutive even integer are

2n , 2n +2 , 2n+4

Their sum 2n + 2n +2 + 2n +4 = 6n + 6 = 12

So 6n = 12 – 6= 6

n = 6/6= 1

three integers are 2n = 2 ,2n+2 = 4, 2n+4 = 6

the integers are 2, 4,, 6

AND A) 6,8,10 B)2,4,6 C) 0,2,4

Correct ANS B)

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9) kx – 19 = k – 1 and 3 – k = 0 ,what is the value of 3x – k ?

From 3 – k = 0 , 3 = k

So k = 3 substitute in kx – 19 k = k – 1

Then x * 3 – 19 * 3 = 3 – 1 ( * is used for multiplication)

Simplify 3x – 57 = 2

3x = 2+ 57 =59

Divide by 3 on both side

3x/3 = 59/3

x = 59/3

2x-k=3*59/3 -k = 59- 3=56 (substitute the value k=3)

Ans A) 56/3 B) 53/3 C) 58/3

Correct ans A

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linear eauations in two variable

Linear eqation with two variable:

Example

1)a + 2b =10 and – a + b =2 solve and find the value of a and b,

a + 2b = 10

-a + b = 2 (add the equations)

3b = 12

Divide by 3 on both sice

3b/3 = 12/3

b = 4

substitute the value of b = 3 in a+ 2b = 10

a + 2 *4 =10

a +8 +10

so a = 10 – 8 =2

Ans : a=2 and b=4

ANS

(2,4) B) (4,2) C) (1,4) D (4,1)

Correct Ans (2,4)

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Consistence of linear equation.

ax + by =c and px +qy = r are system of linear equation

) In the above equation

a/p is not equal to b/q , Then the pair equation are consistent and it has one solution. And dependent

The Ratio of coefficient of the variable X is not equal to the ratio of coefficient y in pair of linear equations

Then they are consistent, has only one solution and dependent

a/p equal to b/q and is not equal to c/r

Then the equations are in-consistence .

The equation are parallel and the equation has no solutions

The ratio of variable x and variable y are equal but the ratio of

Contant are not equal, then they are parallel and incosistant

a/b equal to b/q equal to c/r then the lines are co- incident and they have many solutions

The ratio the variable x and y and the ration of the constant

Are equal then they consident.and they have many solution

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Example

X + y = 5 , 2x + 5y = 19 check the consistence of the equation

Ratio of the coefficient of variable x =1/2

Ratio of the coefficient of the variable y = 1/5

½ is not equal to 1/5

They are cosistant

A ) consistent B )inconsistent c) co-inside

Correct ans Consistent

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2) kx – 3y = 4, 4x – 5y = 7

Where k is a constant and x and y variables

For what value of K ,the system linear equation has no solution

The ratio of the coefficient of the variable x = k/ 4

The ratio of the coefficient of the variable y -3 /- -5 = 3/5

The condition for no solution the coefficient of the variable x and y are not equal

So k/4 is not equal to 3/5

k/4 is not equal to 3/5 (multiply by 20 on both side)

k/4 * 20 is not equal to 3/5 *20

5k is not equal to 12

K is not equal to 12k/5.

Ans k is not equal to 12/5

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Parallel and perpendicular lines:

Example

ax + by = c and px + qy = r

if these two equations are parallel the their slopes are equal

slope of the equation ax + by = c

b y = c -ax ( re arrange)

divide bothside by b

c/b- ax /b, y = -a/bx +nc/b) (rearrange the equation)

slope m =m= -a/b ( slope is equal to the coefficient of x)

slope of the equation px + qy + r

qy= r- px, divide both side by p

qy/q = r/q-px/q

y = -px/q +r/p

slope of this equation is n =-p/ q

if slope m = slope n

Then they are parallel.
the product of the slope is negative one then they are perpendicular

Thai is m *n= -1

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Parallel and perpendicular lines:

Example

ax + by = c and px + qy = r

if these two equations are parallel the their slopes are equal

slope of the equation ax + by = c

by = c -ax ( re arrange)

divide bothside by b

by /b = c/b- ax /b, y = -a/bx +nc/b)

slope m =m -a/b ( slope is equal to the coefficient of x

slope of the equation px + qy + r

qy= r- px, divide both side by p

qy/q = r/q-px/q

y = -px/q +r/p

slope of this equation is n =-p/ q

if slope m = slope n

Then they are parallel.
the product of the slope is negative one then they are perpendicular

Thai is m *n= -1

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2)Line p is defined by the equation. 6y + 12x = 9

Line r is perpendicular to p in xy plane.

What is the slope of line r and write the equation of r if it passes through the point (0 ,4).

Solution

Line r is perpendicular to p.

Therefore slope of p is – 1/m, where m is the slope of line p

The slope of p

6y = 9 – 12 x = -12 x + 9

Divide 6 on both side

6y/6 = -12x/6 + 9/6

Y = -2x +9/6

Solpe m= -2

Line perpedicular to p is -1/m = -1/-2 =1/2

Line r passes through the point ( 0, 4)

The x intercept of the line p is 4

Equation of r is y = mx + b

Y = 1 x/2 + 4

ANS

Y = 1x/2 + 4 B) y = -1x/2 + 4 c) y = 2x + 4 D) y = -2x -4

Correct ANS A

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3) The point s lies on the line y – 3 = -4 ( x + 1 ).

If y co-ordinate of the line s is -1. Then what is the value of x co-ordinate.

Solution

Substitute the y = – 1 in the equation y – 3 = -4 (x + 1)

-1 – 3 = -4 ( x + 1 )

-4 = – 4( x + 1 )

Divide both side br -4

Then -4/-4 = -4 ( x + 1)/-4 =(x + 1 )

= x + 1

X = 0

Ans A) 0 B) 2 c) -1 D ) -2

Correct ans is 0

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Reflection of a line on y axis

If y = mx + b then the reflection this line on y axis is also called flip over.

To get the reflection of line on y axis negative the coffecient of x

Reflection or flip over on y axis ix y = -mx + b.

Example

4) If line y = -3 x + 2 then what is the equation of the reflection of the line y = -3x +2 on y axis.

Solution;

To get the reflection of a line on y axis

Negative the coefficient of X

Therefore equation of the reflection of y = -3x + 2 is

Y = -(-3)x + 2 =3x + 2

ANS A) y = 3x -2 B) y= 3x +2 C) y = 4x -1

D ) y = x -2

Correct ans B

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5)the graph of the equation p x + 5 y = r is a line in XY- plane

Where p and r are constant. If the line contains the points (0, 6) and (-1 ,2) , what is the value of r .?

Solution:

The line px + 5y = r pass through ( 0, 6 )

Then substitute x = 0 and y = 6

0+ 5 * 6 = r

30 = r

ANS A) 20 B ) 30 c) -30 D )35

Correct ans B

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6) X 12 17 24

Y 120 140 150

The table shows the vale of x and y for three point on a

Line l and line p is the translating line of l up by 2 units

In the XY – plane. What is the equation of line p.

Solution:

Let the equation of the line l y =mx + b

Here m and b are constant and m is the slope of the line l

Slope m = y2 – y1/ x2 – x1 ( difference of the co-efficent of y is divided by the difference of the co-effcent of x)

m = 140-120/ 17 – 12 =20/5= 4

substitute the value m and (12, 120 ) in

y = mx + b

120 = 4*12 + b= 48 + b

120 – 48 = b

72 = b

The equation is y = 4x + 72

The translating of l by 2 is

Y = 4x + 72 + 2= 4x + 74

ANS: The translating equation is

Y = 4x + 74

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6) y= mx + b represents a straight line (3 ,7) ,(k,13) and ( -1 , 1)and are the three point on the straight line.

Find the value of k

Solution

Slope of the equation m = y2 – y1/ x2 – x1

For the points (3,7) and ( -1 , 1))

Substitute the co-ordinate of the above two point in slope

m = -1 – 3/ 1-7= -4/-6 = 4/6 =2/3 (dived by 2)

m = 2/3

slope of the line for the point (3 , 7) and ( k ,13)

m= 2/3 = k – 3/ 13 – 3 = k – 3/ 10

2/3 = k -3/10 ( multiplied by 30 ,the least com

Multiple of 3 and 10)

2/3 *30 = ( k -3 )*30/10

2* 10 = ( k- 3) *3

20 = 3k – 9

Re arrange 3k = 20 – 9 K = 11/ 3.

Therefore K = 11/3

ANS A) 11/3 B ) 3/11 C)1/3 D) 3

Correct ANS A

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7) In the fallowing data reperesent the co-ordinated the x and y

X 1 2 3 4

Y 7 10 13 16

Which of the following equation represent the above ordered pairs,

( ordered pair means the co-ordinates of x and y)

Y = 3x + 4 B) y = – 3x + 4 C) 2y = 3x + 1 D) y – 3x + 5

Solution :

HINT : from the table x co- ordinates are incrised by one and the y co- ordinates increased by 3

It means that for every x co– ordinate the value of y co- ordinate increased by 3

The slope of the equation the line m = 3/1

General equation of any straight line with the slope m and intercept form is

Y = m x + b now subtitue the value of m = 3 and the point (1,7)

Y = 7 = 3 x + b

3x + b = 7 re arranged

3 * 1 + b = 7 , 3 + b =7

b = 7 – 3 = 4

therefore equation of the line y = mx + c is

y = 3x + 4

correct ans is A

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6) y= mx + b represents a straight line (3 ,7) ,(k,13) and ( -1 , 1)and are the three point on the straight line.

Find the value of k

Solution

Slope of the equation m = y2 – y1/ x2 – x1

For the points (3,7) and ( -1 , 1))

Substitute the co-ordinate of the above two point in slope

m = -1 – 3/ 1-7= -4/-6 = 4/6 =2/3 (dived by 2)

m = 2/3

slope of the line for the point (3 , 7) and ( k ,13)

m= 2/3 = k – 3/ 13 – 3 = k – 3/ 10

2/3 = k -3/10 ( multiplied by 30 ,the least com

Multiple of 3 and 10)

2/3 *30 = ( k -3 )*30/10

2* 10 = ( k- 3) *3

20 = 3k – 9

Re arrange 3k = 20 – 9 K = 11/ 3.

Therefore K = 11/3

ANS A) 11/3 B ) 3/11 C)1/3 D) 3

Correct ANS A

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7) In the fallowing data reperesent the co-ordinated the x and y

X 1 2 3 4

Y 7 10 13 16

Which of the following equation represent the above ordered pairs,

( ordered pair means the co-ordinates of x and y)

Y = 3x + 4 B) y = – 3x + 4 C) 2y = 3x + 1 D) y – 3x + 5

Solution :

HINT : from the table x co- ordinates are incrised by one and the y co- ordinates increased by 3

It means that for every x co– ordinate the value of y co- ordinate increased by 3

The slope of the equation the line m = 3/1

General equation of any straight line with the slope m and intercept form is

Y = m x + b now subtitue the value of m = 3e and the point (1,7)

Y = 7 = 3 x + b

3x + b = 7 re arranged

3 * 1 + b = 7 , 3 + b =7

b = 7 – 3 = 4

therefore equation of the line y = mx + c is

y = 3x + 4

correct ans is A

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Linear function in Algebra

A linear function is defined by f ( x ) = ( x +2) (x – 3)

Find the value of f (- ! )

Solution:

To find the value of the function f ( – 1 )

Substitute x = -1 in f (x ) = ( x + 2 ) ( x – 3 )

f ( – 1 ) = ( -1 +2 ) ( -1 – 3 ) =1 ( – 4 )

= -4

ANS A) 4, B ) – 4 , C) 5 , D ) 6

Corrcet ANS B

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2)Distance traveled by a an object is given by d = 6 t,

Where t is the time , explain the constant 6

Solution

Distance of an object = speed * time

Here the given equation is d = 6 t

Where t is thime

So 6 repesent the speed of the Object.

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4)A linear equation is given by f (x ) = 20

Which of the fallowing table satisfy the above equation

X 0 1 2 B) x 2 3 5 C) x 10 20 30 D) x 5 6 7

Y 20 20 20 y 0 26 45 y 20 15 20 y 20 19 30

Correct ANS A

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5) A function is P is defined as p ( x ) = – 2 x

Find for what value of x p ( x ) = -24

Solution :

To find the value of x

Equate p ( x ) = – 24

Then p ( x ) = -24 = – 2 x

Divide both side by -2

Therefore – 2 x /-2 = – 24 /-2

X = 12

ANS A ) 10 , B ) -10 , C ) 12 , D ) -12

Correct ANS C

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A sample model taken from a farm the new born horse has a weight 120 pounds and its weight is

Increased by 2 pounds per day .

Find the weight of the Baby horse after a week

Solution

From the given data we write

Weight of the Baby horse after a week = weight of the horse at the gtime of born

+ 2 * 7

= 120 + 14 =134 pounds

ANS A) !35 , B) 136 , C) 137 D) 134.

Correct ANS D

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Inequalities of linear equation with two or one variable

Which of the following following ordered pairs satisfy the linear inequality equation

Y > = 2 x – 1 and y < x + 1

(0,0) B (1,1) C (-1, -1)

Solution:

Substitute (0,0 )

In y > = 2 x – 1 ( .> = means greater and equal) and y < x + 1

0 > = – 1 true and 0 <0 + 1 , 0 < 1 true

Substitute (1, 1) and 1<1+1

Y > = 2*1 – 1 1<2 true

= 2 – 1
= 1

1>=1 false

(1 , 1 ) false

Substitute ( -1 , – ! )

-1 >= 2* -1 – 1 and -1< -1 + 1

–1 > = – 2 – 1 -1 < 0

-1 > = -3

True true

ANS correct answer A and C

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2) A coffee shop selle coffee and tee to atleast $ 3000 per day

The cost each unit of coffee is $4 and per unit tee cost $3 .

Which one the fallowinh represent the per day sale the shope.

Solution

Let Number of coffee and tee sold per day are c and t

The cost of coffee and tee per unit price are $4 and $3

The equation is

4 c + 3 t > = 3000.

ANS

4c + 3t > = 3000

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3) A Boater rent his boat at the tare of $ 55 per hour, he charged $ 10 as his service charge,

He wants to get a minimum imcome 1000 per day.

Which one is correct form of repesention

10 + 50 h >= 1000 B) 10 + 50 h <=1000 c) 10 + 55h >1000 D) 10 + 50h < 1000

Solution

Service fee $ 10

Rent per hour =$ 50

Nomewr of hours rented be h

Minimum income1000

Equation for this 10+ 50h >= 1000

Correct ANS A

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hello srija

Linear function problems with graph
10×10 Grid with Line y = 3x + 5

4)The graph is shown in above diagram is in XY-plane.

is represents the distance travelled by a moving object in a Stright line With respect to tine in miles

what is the speed of the moving object

Option

A)2miles per unit time
B) 2.5 miles per unit time

C)4 miles per unit time

D)1.5 miles per unit time

definition of speed : Distance travelled by an moving object in unit time is called speed.

here the distance travelled by a moving object is represented by a Stright line

so here the speed of the object is given by the slope of the stright line

formula for the slope of Stright line m= y2-y1/x2-x1 ,where (x1,y1) and (x2, y2) are co-ordinates

of any two points on the Stright line graph

here the co-ordinates of the points on the stright line graph are (1,0), (2,2), (3,4), (4,6), (5,8)

Consider any two points, here i am considering (2,2) and (5,8)

slope m = y2-y1/x2-x1=8-2/5-2=6/3=2

The speed is = 2 miles per unit time

Correct ANS:A

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Hello pathu

Helo pathu

Hello pathu

The above equation represent the cost os a certain pen with respect to time.

what is the equation of the graph

Option

A)y=2x-3

B)y=2x+3

C)y=3x-2

D)y=3x+2

solution

General equation of a Stright line y = mx +c

m=y2-y1/x2-x1

where m-is the slope of the equation and c - y intercept

consider the point (0,-3) and (1.5,0)

substitute the value of x and y co-ordinates from the above points

m= 0- (-3)/1.5=0

m= 3/1.5=, Multiply by 10 both numerator and denominator

m= 3*10/1.5*10=30/15=2

therefore the equation y= 2x +c

to find the value of c ,consider the point (0,-3)

x=0 and y=-3

-3=2*0 +c

-3= c

substitute the value of c= -3

y = 2x -3

correct ANS:A

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Hello poorvaja

hello poorvaja

in the graph represent a function f(x)= 2x+k

Find the value of K

It is given that the graph represent a function f(x)=y=2x+k

the graph passes through the point (1,6)

so it satisfy the equation of the function

That is f(x)=y=2x+k

Therefore substitute x=1 and y=6 in the above equation

y=2x+k, y=2*1+k=6

2+k=6

k=6-2

k=4

The correct ANS:4

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7)How many points the given two graph intersects with each other in XY-plane.

Y=-x+10 and y=5x.

Option:

A) 0ne point

B)two points

C) no point

D)many points

Solution:

Each of the equation is written in slope and intercept form

That is in general form of intercept and slope form

Y=mx+C.

Compare the given equation y=-x+10

Here the slope m1=-1 and intercept is 10

Now compare with the equation y=5x

Here the slope m2=5 and intercept is 0

Therefore the above equation has different slope

That is m1≠m2

They are consistent

The lines are intersect at only one point.

Corrcet ANS:A

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8)The equation g(x) is defined as g(x)=5e^x-4 in XY- plane .Find the x- intercept of x and y.

Option:

A) 0 and 1

B) 0 and 0

C)1 and 0 D)

Solution:

Let The given equation g(x)= y=5e^x-5

To find x- intercepts substitute y=0

Therefore 5e^x-5=0

5e^x-5=0 (By addind 5 on both side of the equation )

5e^x-5+5=5

5e^x=5 ( divide both side of the equation 5)

Now the equation 5e^x/5=5/5=1

e^x=1 (express 1 in exaptational form)

e^x=1=e⁰

Therefore x – intercept is 0

To find Y- intercepts substitute x=0

In the above equation y=5e^x-5

Therefore y=5e^x-5 .

Y= 5e^x-5 . = 5e⁰-5

5-5=0

therefore y intercept is 0

Correct ANS:B

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9)The function f(x) is defined as f(x)=〖(x-1)〗²+〖(x+1)〗²-10 in XY-plane.

The points (0,0) and (t,0) lies on the above graph of the function.

Find the positive value of t.

Option :

A)2

B) -2

c)3

D)-3

Solution:

Conside the equation f(x) =y =(x-1)²+(x+1)²-10 ,

Now substitutes (t,0) in the equation.

That is x=t and y=0 in the above equation

(t-1)²+(t+1)²-10

t²-2t+1+t²+2t+1-10=0

2t²+2-10=0

2t²=8=0

Divide both side by 2

2t²/2-8/2=0,/p>

t²-4=0

t²=4

t=√4=+2 or -2

Correct ANS:A

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Problems with difficulty level-high

System of two linear equations in two variable:

1)Solve the following equation for the value of n under condition

the given system of linear equations has no solution

3+5x=ny and 5 x-3y=3y+15

IOption:

A)6

B)5

C)-6

D)-5

Spolution:

From the given equation 3+5x=ny

Add -3 on both side

Therefore 3+5x-3=ny-3

5x=ny-3

Substitute the value of 5x in equation

5x-3y=3y+15

So we will get ny-3-3y=3y+15

Add 3y on both side

Now ny-3-3y+3y=3y+3y+15

ny-3=6y+15

Now apply the system of linear equation has no solution

This gives that ny-3≠6y+15

This is true if ny=6y

So from this we can conclude that n=6

Correct ANS:A

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2)A stationary store selling two types of pens at the rate of $15 and $20 each.

The store sold 35 pens for $600.Find the number of pens sold in each kind.

Option:

A)15,20

B)20,15

C)20,13

D)20,15

Solution:

Let the number of pens sold in each kind are a and b.

Total number sold =35

That is a+b=35.

Tolat cost of sale =600

That is the total cost of a and pens

That is 20a + 15b=600 ( cost pens 420, 15 -given)

consider a+b=35

add -b on both side

a+b-b=35-b

Now a =35-b

Substitute in 20a+15b=600

Then we get 20(35-b)+15b=600

Simply 700-20b+15b=600

Add -700 on both side

700-20b+15b-700=600-700

-20b+15b=-100

-5b=-100

Divide -5 on both side

-5b/-5=-100/16

b=20

Substitute in b= 20 in a+b=35

Then a+20=35

Add -20 on both side

Now a+20-20=35-20

a=15

Correct ANS:A

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3)Find the value of p and q for the from the following equation

And the equation have no solution

3/2x-5/3y =7/9 and px + qy =r , where p ,q and r are constant.

Option:

A)3/2,-5/3

2)-3/2,5/3

C)5/3.-3/2

D)-5/3,3/2

Solution:

3/2x-5/3y=7/9

Px=qy=r

From the above equation

The ratio of x coordinates=3/2/p

The ratio of y co-ordidates =-5/3/q

It is given that these equation have no roots

Therefore 3/2/p=-5/3/q

Therefore p=3/2 and q=-5/2

Correct ANS:A

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4)A rope of 250 meters long .cut in to two parts .one part is 10 meter less then 5 times of the other part.

What is length of larger part of the rope in meters.?

Option:

A)240

B)250

C)120

D)180

Solution:

Let p and q are two part of the rope.

P + q = 250. (length of the rope,

Let p the smallest part of the rope.

And q is larger part of the rope.

It is given one part of the rope 10 meters less than the 5 times of the other

Therefore q -10=5p

Add 10 on side

Then q-10 +10=5p+10

q=5p +10

Substitute q value in p+q = 260

Therefore p+5p+10=250

add =10 on both side of the equation

6p+10-10=250-10

6p=240

Divide by 10 on both side

6p/6=240/6

P=40

The other part of the rope q =250-10=240 meters

Correct ANS: A

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5)Solve the following system of linear equation and find the value of √6xy

The equations are 3x+7y=23 and 7x-3y=15.

Option:

A)6

B)9

C)5

D)7

The equations are

3x+7y=23

7x-3y=15

Now consider 3x+7y=23

Add -7y on both side of the equation

Therefore 3x+7y-7y=23-7y

That is 3x=23-7y

Divide 3 0n b0th side

3x/3= (23-7y)/3

X=(23-7y)/3

Substitute the value of x in 7x-3y=15

That is 7((23-7y)/3)-3y=15

Multiply by 3 on both side

7((23-7y)/3)×3-3y×3=15×3

7(23-7y)-9y=45

161-49y-9y=45

161-58y=45

Add -161 on both side.

161-58y-161=45-161

-58y=-116

Divide by -58 on both side

(-58y)/(-58) =(-116)/(-58)

Y=2

Substitute y =2 in 3x+7y=23

3x+7×2=23

3x+14=23

Add -14 on both side

3x+14-14=23-14

3x= 9

Divide 3 on both side of the equation

3x/3=9/3

X=3

Now substitute the value x=3 and y=2 in √6xy =√(6×3×2)=√36=6

Correct ANS;A

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6)A teacher order maths text book and science textbook for his class on the total he order a 30 text books.

He spend a sum of$ 230 for the 30 textbooks. And cost of a

math textbook $15 each and q science textbook $ 10.

solution:

Let the number of math's textbook ordered =m

and science textbook =s

total number of books ordered =30

therefore m+s=30

Now the cost of each maths text book =15 and science textbook = 10

Total cost =230

Therefore 15m+10s=230

The equation for given statement is m+s=30 and 15m+10s=230

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Find the value x and y in the following system of linear equations.

3x+5y=11 and 4(3x+5y) =46.

Option:

A)(2,1)

B)(2,1)

C) (2,3)

D) (3,2)

Solution:

Consider the given equation

3x+5y=11 and 4(3x+5y)=46

Now substitute the value of 3x+5y = 11 in equation 4(3x+5y)+2y =46

Therefore 4(11) +2y=46

44+2y=45

Add -44 on both sides of the equation

44+2y-44=46-44

2y=2

Divide by 2 on both side

2y/2 =2/2

Y=1

substitute the value of y=1 in

3x+5y=11

3x+5x1=11

3x+5=11

add -5 on both side of the equation

3x+5-5=11-5

3x=6

divide 3 on both side of the equation

3x/3=6/3

x=2

therefore the value of x=2 and y=1

Correct ANS :A

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8)Discus the consistence of the following equation

3x+2y=21 and 2x+y=12.

Solution

the consistence of the general equation of a system of linear

Equation ax+by=c and px+qy=r is

The ratio of the co-coeffect x is not equal to the ratio of the co-efficient of y

Then the system of linear equation are consistent

Otherwise they are in-consistent

Consider the given equation

3x+2y=21 and 2x+y=12

The ratio of the x- coefficient=3/2

The ratio of y coefficient =2/1

Here 3/2≠2/1

The equations are consistent

Correct ANS: The equations are consistent

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8)Find out for the equation 3x + 5y =29 has no solution with given equation.

Option:A)–19x-15y=29

B)29x-15y=35

C)9x+15y=38

D)-9x-15y=-87.

Solution:

We know the general rule for no solution for a system of linear equation

ax+by=c

and dx+ey=f

is a/d=b/e≠ c/f

That is the ratio of coefficient is equal to the ratio of the coefficient of y and x not equal to the

coefficient of constant

Here consider the equation 3x+5y=29 and take any one of the equations given option

Here consider the condition the ratio of the coefficient of x and y are equal.

The first equation is not valid this condition

-19/3 and -15/3 are not equal

For the second equation

The ratio of x and y are

29÷3 and -15÷5 is not equal

For the third equation

9÷3 and 15÷5 both are 3 they are equal

Consider the last one

The x and y ratio are

-9÷3 and -15÷3 are equals,/p>

Now consider the ratio of their constant

That is - 87÷29=-3

Here the ratio of the coefficient of x and y are equal in the last two.

But in the last one we will see that their constant is also equal

So, this not satisfying our condition that the system equation have no solution.

So, the third equation

9x +15y = 38 is satisfying our condition that the system of equation has no solution

Correct ANS: C

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10)For the system of linear equation 5x + 6y =16 and 10x = 12y =32 has many solution

Which of the following point satisfy the above equation.?

Where r is a constant.

Option:

A)( r+ 2, (-5r)/6+2)

B) ( 2 -r, 5/6 r+1)

C)(5+r,(-5)/6r+2)

D)( r-5,5/66+1)

Solution:

Consider the system of equation 5x + 6y =16 and 10x + 12Y = 32

We can understand that the first equation is multiply by 2 we got the second equation.

So that it is enough given p≠oint can stisfy one of the equation

Consider the point (r+2, (-5r)/6+2)

Substitute x= r+2 and y=(-5r)/6+2

Now 10x + 12 y =32 is equal to 10 (2+r) + 12( (-5r)/6+2)=32

Simplify 20+_10 r +12((-5r)/6)+12×2=32

20+10r + 2(-5r)+24=32

20+10r-10r+24=32

34 ≠32

Consider the point (2-r, r, 5/6 r+1)

Substitute x = 2-r and y = 5/6 r+1

10x + 12y =32

10(2 – r ) + 12 ( 5/6 r+1)=32

20 – 10r + 12 X 5/6 r+1x12=32

20– 10r + 10 r + 12=32

20+12 = 32, it is true

It is the point liues on both equation

You may check for other points also.

Correct ANS: B

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11) In a cricket match the winning team scored 10 runs more three times the losing team.

Find the runs score by the winning

team if the total number of runs scored by both team is 450.

Option;

A)300

B)440

C)340

D)240

Solution:

Let x be the runs scored by losing team

And y be the 10 more than three times the runs scored by losing time.

That is y=10+3x

Total number of runs scored by the both team is 450

So x +10 + 3x = 450

4x+10=450

Add -10 on both side

Then 4x+10-10=450-10

4x=440

Divide 4 on both side

4x/4=440/4

X=110

Runs scored by winning team = 3x+10=3x110+10=330+10=340.

Correct ANS:340

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Linear equations in one and two variable

1)If p(5x-25) = x-5 then find the value of p where p is a constant

And the given equation has many solution,

Solution:

Consider the given equation p(5x-25)=x-5.

Simply 5p(x-5)= (x-5)

Divide both side by x-5

Then (5p(x-5))/(x-5)=(x-5)/(x-5)

5p=1, divide by 5 on both side of the equation

5p/5=1/5 , p=1/5

When p is 1/5 then both side of the equation are equal

So it has many solution

Correct ANS :1/5

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2) Find the value of a , when the following equation has no solution

(5-a)x -15=2ax+5

A)5/2

B-5/2

C)<2/5

D)-2/5

Solution:

Consider the given equation (5-a) x =2ax+5

Simplify 5x-ax-15 =2ax +5

Add -2ax on both sides then

5x-ax-2ax -15=2ax+5-2ax

5x-3ax-15=5

Add 15 on both side

Then 5x-3ax -15+15=5+15

5x-3ax=20

Simplify x(5-2a)=20

Divide (5-2a) on both side

Then (x(5-2a))/(5-2a) =20/(5-2a)

X=20/(5-2a)

In this equation if 2a =5 the the right side of the equation is undefine

So it has no solution

That is when 2a=5

It gives a=5/2

Correct ANS:A

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3) discuss about the solution of the following equation

5(21y-6) = 15/2(14x-6).

Solutiuon: Consider the given equation

5(21y-6)=15/2(14y-6)

Multiply by 2 on both side

5(21y-6) ×2=15/2(14y-6)×2

10(21y-6)=15(14y-6)

Simplify

10x3(7y-2)=15x2(7y-3)

30(7y-2)=30(7y-3)

Both side of the equation are equal

So the given equation has many solution.

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4)The line k has a slope 3/2 and it has x – intercept of (-3,0)

On XY-plane. Find the y coordinate of y intercept,

Option:

A)-9/2

B)9/2

C)2/9

D)-2/9

Solution:

Consider the slope m=3/2 and coordinates of x-intercept (-3,0)

Equation of a line with m and y intercept is = mx + c

M- slope ,c- y intercept

here m=3/2

therefore y=3/2 (x+c)

coordinate of x intercept_-3,0)

substitute in the equation y=3/2(x+c)

x=-3 and y=0

then we get 0=3/2(x-3+c)

-9/2+c=0

Add 9/2 on both side of the equation

-9/2 +c=0+9/2

C=9/2

The y coordinate of y intercept is 9/2

Correct ANS:B

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5In the graph 9x-5y=-12 in the XY-plane. Has an x-intercept at (p,0) and a intercept at (0,q)

Where p and q are constant.

Findt value of pq .

Option:

A)38/15

B)15/38

C)-38/15

D)-15/38

Solution:

Consider the given equation 9x+5y= -12.

Substitute the for x= p and y=0 for the point ( 0 ,P)

9p=-12

Divide by 9 on both side

9p/9=-12/9

P=-12/8

Consider the point (0,q)

Substitute x=0 and y=q

5q=-12

Divide by 5 on both side

5q/5==12/5

Calculate the vale for pq=-12/9x-12/5=144/45

Simplify

e get 38/15

Correct ANS:A

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