# Linear Equations

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Linear Equations:

What are Linear Equations?
An equation is a statement of equality of two algebraic polynomial involving one or more variables. The expression of the form Ax + B, where A and B are real numbers and A ≠ 0, is a linear polynomial and equation involving only linear polynomial are called linear equations.

Rules for solving linear equation

• If same number is added to both the sides of an equation, the equality remains same.
• If same number is subtracted to both the sides of an equation, the equality remains the same.
• If same is multiplied to both the sides of the equation, the equality remains the same.
• If both sides are divided by some non zero number the equality remains the same.

Algebraic Methods of Solutions

Substitution Method –

• From either equation find the value of one of the unknown in terms of the other.
• Substitute the value thus found in the other equation.
• Solve the resulting equation involving only one unknown.
• Substitute the value of this unknown in the equation obtained in step first to find the other unknown.

Elimination Method –

• Multiply the coefficients of the equations with some constant so as to make the coefficients of one of the variables to be equal to the coefficients of same variable in other equation.
• If the co-efficient of x or y have the same sign then subtract, if opposite signs, then add the resulting equations.
• The resulting equation will have only one unknown y or x and can be solved easily.
• Substitute value of unknown found in above step in either of the equation and find the other unknown.

Method of Comparison –

• From each of the comparison, find the value of one of the variables in terms of the other.
• Equate the results, solve the resulting equation.
• Substitute the values in either of the results obtained is first step and find the value of other variable.

Cross Multiplication

# Aptitude Questions (Linear Equations):

1. Solve: 2(x – 3) – (5 – 3x) = 3 (x + 1) – 4(2 + x)

A)    4

B)    1

C)    3

D)    2

E)    None of these

Explanation:
2(x – 3) – (5 – 3x) = 3 (x + 1) – 4(2 + x)

=>2x – 6 – 5 + 3x = 3x + 3 – 8 – 4x

=>5x – 11 = -x – 5

=>6x = 6 => x = 1

2. The length of a rectangle is 8cm more than its breadth. If the perimeter of the rectangle is 68 cm, find its length and breadth.

A)    21cm

B)    14cm

C)    18cm

D)    20cm

E)    None of these

Explanation:
Let the breadth of the rectangle be x.
Then, its length = (x + 8) cm
∴ Perimeter of rectangle = 2 [x + (x + 8)] = 2[2x + 8] = 4x + 16
∴ 4x + 16 = 68
=>4x = 68 – 16 = 52

=>x = 13

=>Breadth of rectangle = 13cm and length = 13 + 8 = 21cm

3. A man when asked how many hens and buffaloes he has told that his animals have 120 eyes and 180 legs. How many hens have he?

A)    20

B)    40

C)    30

D)    10

E)    None of these

Explanation:
Let number of buffaloes = x
The number of hens = y
∴ Total eyes = 2x + 2y = 120
∴ Total legs = 4x + 2y = 180
Subtracting,

2x + 2y = 120
4x + 2y = 180
–       –         –
—————-
-2x = – 60

=> x = 30
Put (i) 60 + 2y = 120 => 2y = 60
y = 30
Hence, number of hens = 30

4. If 25x – 19 – [3 – (4x – 5)] = 3x – (6x – 5), x is equal to

A)    x =1

B)    x = -1

C)    x = ½

D)    x = 2

E)    None of these

Explanation:
=>25x – 19 – [3 – (4x – 5)] = 3x – (6x – 5)
=>25x – 19 – [3 – 4x + 5] = 3x – 6x + 5
=>25x – 19 + 4x – 8 = -3x + 5
=>29x + 3x = 5 + 27
=>32x = 32
=>x = 1

5. If (x2-3x+2)/(x2-5x+4) = (x2-6x+8)/(x2-9x+14), then the value of x is

A)    2(1/2)

B)    1/2

C)    2

D)    -2

E)    None of these

Explanation:
(x-2)(x-1)/(x-4)(x-1) = (x-2)(x-4)/(x-2)(x-7)
=>x-2/x-4= x-4/x-7
=>x2 – 9x + 14 = x2 – 8x + 16
=>x = -2