Practice Examples – Linear Equations

#Algebra

Review

Practice Examples

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Q.1

Solve the following equation and find the value of   y?

y – ( 4 – y ) = 2

A.  9

B.  3

C. 2

D. 7

E. 8

Solution

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Correct answer choice is B.

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Q.2

Quantity A                                              Quantity B

                12(2x – 1 )  = 0                                   6(4)x-6-6 = 0

A.  If Quantity A is greater

B.  If Quantity B is greater

C.  If Quantity A and Quantity B are equal

D.  The relationship cannot be determined from given information

Solution

Let us solve the equation for Quantity A

12(2x-1)  = 0

24x-12 = 0

24x = 12

x = 1/2

Now we will solve the equation for Quantity B

6(4) x – 6 – 6 = 0

24x -12 = 0

24x =12

x =12/24

x = 1/2

If we compare Quantity A with Quantity B, both are equal thus, correct answer choice is C.

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Q.3          

Solve the following equation form and n.

2m+ 2m – n = 6

m + n = 4

A.  m =2 , n = 2

B.  m = -2, n = 2

C.  m= 4 , n = 0

D.  m = 8 , n = -2

E.  m= 3 , n = 3

Solution

Let us solve this problem using substitution method

Let us solve this problem using substitution method

4m  – n  = 6  ———— Equation 1

m + n =4 ———— Equation 2

Taking the value of    m   from equation 2

 m  =  4 –n

Substitute this value of m in equation 1 we will get;

4 ( 4 -n)-n= 6

16 – 4n-n = 6

-5n = 6-16

-5n = -10

    n = 10/5

     n = 2

Now we can easily find the value of  m  by substituting the value of  n  in equation 1

4m  – n  = 6

4m = 6 + n

4m = 6 + 2

m = 8/4

 m = 2

 Correct answer choice is A.

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Q.4

Find the value of  x which satisfies following equation.

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                   3(11x – 5 )  =  3x – 4

33x – 15  = 3x – 4

33x  –  3x  = 15 – 4

30x = 11

           x = 11/30

          Correct answer choice is D.

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Q.5

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A.  If Quantity A is greater

B.  If Quantity B is greater

C.  If Quantity A and Quantity B are equal

D.  The relationship cannot be determined from given information

Solution

Let us solve the equation for Quantity A

 (10x- 2)/3   + 3x =   5/2

Multiply both side by 2

20x – 4 + 18x = 15

38 x = 19

x =19/38

x =1/2

Now we will solve equation for Quantity B

( 4x + 48- 42)/12    = 1

4x + 48 – 42 = 12

4x = 12 – 6

 4x = 6

x=6/4

x =3/2

If we compare Quantity A with Quantity B, The value of for quantity B is greater, hence correct answer choice is B.

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Q.6

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= -3/4

 Hence, correct answer choice is ‘D’.

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Q.7

Quantity A                                       Quantity B

5x- 1 = 5(x + 8)                                   2

A.  If Quantity A is greater

B.  If Quantity B is greater

C.  If Quantity A and Quantity B are equal

D.  The relationship cannot be determined from given information

Solution

First we will solve the equation for Quantity A

5x- 1 = 5(x + 8)

5x   – 1 =    5x + 40

5x will be cut both sides so;

-1 = 40. Here we are not able to arrive at any solution.  Thus, Quantity A cannot be compared to Quantity B. Correct answer choice is D.

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Q.8

Solve the following equation for the value of m?

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Cross multiply both sides;

121( m-5)   = 144(6-m-2)

121m – 605  =   576 –144 m

121m +  144m = 576 + 605

 265m = 1181

m =1181/265

                m = @ 4.45

This value is lesser than 5, thus correct answer is ‘B’.

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Q.9

Find the values of m and n?

5m + 6n + 2 = 9

6n – 2m   = 6

m = ____   , n = ____?

Solution

5m + 6n + 2 = 9

6n – 2m   = 6

We will solve this equation using elimination method

5m + 6n = 7

6n – 2m = 6

5m + 6n = 9 ———- Equation 1

-2m + 6n = 6 ——— Equation 2

To make variable m equivalent, multiply both the equations with coefficients of m  by interchanging each other;

5m + 6n = 9        x 2

-2m + 6n = 6      x 5

10m + 12n  = 18

-10m+ 30n  = 30

Add both the equations

 10m + 12n = 18

-10m+ 30n = 30

        42n = 48

         n =48/42

            =    8/7

Subsitute the value of n in equation 1 to get the value of m

5m + 6n = 7

5m + 6 x 8/7= 7

5m = 7-48/7

 5m = 49 -48/7

  5m =1/7

m =1/35

The answer is  m =1/35  and n =  8/7

Let us cross check these values

5m + 6n = 7

5  x  1/( 35)  + 6 x  8/( 7)      =7

1/7+48/7       =7

49/7    =7

7 = 7

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Q.10

From the following options which are not a solution of the equation

5m + 5n = 2n – 4m + 15?

A.  m = 1 , n = 2

B.  m = -1 , n = 8

C.  m = 2 , n = -1

D.  m = 3 , n = -4

E.  m = -2 , n = 9

Solution

  5m + 5n = 2n – 4m + 15

  5m + 4m   + 5n – 2n = 15

  9m + 3n = 15

Solve this sum by putting answers in the equations

Let us start substituting the values given in the option choices

  • If m = 1,   n = 2  ;      9  x  1  + 3 x 2         =  15
  • If m = -1,  n = 8 ;     9 x (-1 ) + 3 x 8   =  15
  • If m = 2,   n = -1  ;    9 x 2  + 3 x (-1)  =  15
  • If m = 3,   n = -4  ;    9 x 3  + 3 x (- 4)  =  15
  • If m = -2,  n = 9 ;      9 x (-2 )+ 3 x 9   =     9

So, correct answer choice is E.

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Q.11

m + 8 = n

(m-3) (n- 1) = 0

  Quantity A                                               Quantity B

       2n                                                            3m

A.  If Quantity A is greater

B.  If Quantity B is greater

C.  If Quantity A and Quantity B are equal

D.  The relationship cannot be determined from given information

 Solution;

   m + 8 = n   ……………….Equation 1

(m-3) (n- 1) = 0………………..Equation 2

In this equation 2 product of two variables is ‘0’ that means either of the variables or both variables can be zero.

So, m-3 = 0 or n -1 = 0

       m = 3 or n = 1

 Condition 1.If m =3, then 3+ 8 = n

n = 11

2n = 2x 11 = 22, 3 x m = 3 x3 = 9, Quantity A is greater.

Condition 2.If n = 1, then m + 8 = 1, m = 1 – 8 = – 7

m = -7

3m = 3 x -7 = -21, 2n = 2 x 1 = 2, Quantity A is greater.

In both the condition, Quantity A is greater. Correct answer choice is ‘A’.

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Q.12

                             m ( n-5 ) = 0

                             m + n = 15

                       Quantity A                                   Quantity B

4m                                            n

A.  If Quantity A is greater

B.  If Quantity B is greater

C.  If Quantity A and Quantity B are equal

D.  The relationship cannot be determined from given information

Solution:

( m + 1) ( n-5 ) = 0………………Equation 1

m + n = 15 …………………Equation 2

In equation 1, multiplication of two variables is zero which is possible only, if m is zero or n is zero or mn is zero.

(m + 1) (n – 5) = 0

m = 0 or n-5 = 0

m = 0 or n = 5

Condition 1; If m = 0 then   m + n = 15 will be

0 + n = 15

n = 15

Quantity A:   m= 0, 4m = 4 x 0 = 0

Quantity B:  n = 15

n > 4m,   Quantity B is greater

Condition 2; If n = 5 then m + n = 15 will be

m + 5 = 15

m = 10

Quantity A: 4m = 4 x m = 4 x 10 = 40

Quantity B:  n = 5

4m > n, Quantity A is greater

One time n is greater and 2nd time 4m is greater, hence the relationship cannot be determined from given information.

The correct answer is ‘D’.

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