**SSC CHSL Topic Wise Study Material – Quantitative Aptitude – Linear Equations**

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An equation is an expression of the structure A = B, where, A and B are expressions containing one or unknowns called variables.

Equations are classified on the basis of their degree i.e., the highest power of the whole expression and the number of variables used in the equations.

Classification of equations on the basis of number of variables used in the equation

**Linear Equation in One Variable**

A linear equation in one variable is an equation of the type ax + b = 0 or ax = c, where a, b and c are constants (real numbers),

a ≠ 0 and x is an unknown variable.

The solution of the linear equation ax + b = 0 is x = – b/a. We also say that – b/a is the root of the linear equation ax + b = 0

**e.g.,** The equation 2x + 3 = 0 is a linear equation in one unknown variable x. Its solution or root is – 3/2

**Linear Equation in Two Variables**

A linear equation in two variables is an equation of the type ax + by + c = 0 or ax + by = d, where a, b, c and d are constants, a≠0, b≠0.

**e.g.,** 3x + 4y + 7 = 0 and 2x – 3y = 5 are linear equations in two variables x and y.

**Methods of Solving Two Simultaneous Linear Equations**

**Method of Substitution**

**Step I** Find the value of one variable, say y, in terms of the other i.e.,x from either equation.

**Step II** Substitute the value of y so obtained in the other equation. Thus, we get an equation in only one variable x.

**Step III** Solve this equation for x.

**Step IV** Substitute the value of x, thus obtained, in Step I and find the value of y.

**Example Solve x + y = 7, 3x – 2y – 11. Find the values of x and y ?**

(a)x=5,y = 2

(b)x=4,y = 3

(c)x=2, y = 5

(d)x=-2, y=-3

**Method of Elimination**

**Step I** Multiply both the equations by such numbers so as to make the coefficients of one of the two unknowns numerically the same.

**Step II** Add or subtract the two questions to get an equation containing only one unknown. Solve this equation to get the value of the unknown.

**Step III** Substitute the value of the unknown in either of the two original equations. By solving that the value of the other unknown is obtained.

**Example Solve -6x + 5y = 2, -5x + 6y = 9. Find the values of x and y.**

(a)x=3, y = 3

(b)x=4,y = 3

(c)x= 3, y = 4

(d) x= 4, y = 4

**Example Solve 3x + 2y = -25, -2x – y = 10. Find the values of x and y.**

(a)x=5, y = 20

(b) x= -5, y = -20

(c) x = -5, y = 20

(d) x = 5, y = -20

**Consistency of the System of Linear Equations**

When a system of equations has a solution, the system is called consistent. When a system of equations has no solution, the system is called inconsistent.

**Test for Consistency**

If we are given two linear equations

a_{1}x + b_{1}y = c_{1} and a_{2}x + b_{2}y = c_{2} Then,

• If a_{1}/a_{2} = b_{1}/b_{2}, the system will have exactly one solution and will be consistent.

Note The graphs of such equations will have intersecting lines.

• If a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2} the system is consistent and has infinitely many solutions.

Note The graphs of such equations will have coincident lines.

• if a_{1}/a_{2} = b_{1}/b_{2} ≠ c_{1}/c_{2}, the system has no solution and is inconsistent.

Note The graphs of such equations will have parallel lines.

**Example For what value of k will the system of equations kx + 2y= 5 and 3x + y = 1 have a unique solution?**

(a)k≠2

(b)k≠3

(c)k≠5

(d)k≠6

**Example For what value of k, the system of equations 3x + 4y = 6 and 6x + 8y = k represent, coincident lines?**

(a) k = 4

(b) k= 8

(c) k= 12

(d) k = 6

**Example For what value of k the equations 9x + 4y = 9 and 7x + ky= 5 have no solution?**

(a)k=27/7

(b)k=28/9

(c)k=29/9

(d )k=29/7

**Important Types of Equations**

1. Linear Equations

2. Quadratic Equations

**In Equations**

In equation is a statement of mathematics usually written in the form of a pair of expressions denoting the values in questions with a relational sign between them indicating the specific inequality relation like a<b,x + y + z≤1 etc.

**Example If p = 3/5, q = 7/9, r = 5/7, then which of the following inequality is true? SSC (10+2) 2012**

(a)p<q<r

(b)q<r<p

(c)p<r<q

(d)r<q<p

**Example If 7n + 9> 100 and n is an integer, then smallest possible value of n is SSC (10+2) 2011**

(a) 13

(b) 12

(c) 14

(d) 15

**Reference Corner**

**1. If the graphs of 3x – 5y = -8 and 3x + 5y = 32 intersect at the point (p, q), then the value of p – q is .SSC (10+ 2) 2013**

(a) 3

(b)2

(c) 1

(d) 0

**2. If 6x -5y = 13,7x + 2y = 23, then 11x + 18y is equal to SSC (10 + 2) 2013**

(a) 33

(b) 15

(c)-15

(d) 51

**Practice Exercise **

**1. One says, “give me a hundred friend! I shall then become twice as rich as you”. The other replies, “If you give me ten, I shall be six times as rich as you”. Find the amount of their capitals.**

(a)Rs 40,Rs 170

(b)Rs 60,Rs 180

(c)Rs 80,Rs 200

(d) Couldn’t be. determined

**2. Renu’s mother was three times as old as Renu 5 yr ago. After 5 yr, she will be twice as old as Renu. Renu’s present age (in yr) is**

(a) 35

(b) 10

(c) 20

(d) 15

**3. The ratio of incomes of two persons 8 : 5 and the ratio of their expenditure is 2 : 1. If each of them manages of save Rs 1000 per month, find the difference of their their monthly income.**

(a) Rs 2500

(b) Rs 1500

(c) Rs 1000

(d) Rs 700

**4. A fraction becomes 7/8, if 5 is added to both the numerator and the denominator. If 3 is added to both the numerator and the denominator, it becomes 6/7. Find the fraction.**

(a)8/11

(b)9/11

(c)10/11

(d)Couldn’t be determined

**5. Father is aged three times more his son Remu. After 8 yr, he would be 2 1/2 times of Ramu’s age. After further 8 yr, how many times would he be of Ramu’s age?**

(a) 2 times

(b) 2 1/2 times

(c) 2 3/4 times

(d) 3 times

**6. For what values of k will the following pair of linear equations have infinitely many solutions?**

**kx+3y-(k-3)=0**

**12x + ky – k = 0**

(a) 2

(b) 4

(c) 6

(d)

**7. 3x² – 7x + 4 ≤ 0**

(a) x > 0

(b) x< 0

(c) All x are zero

(d) None of these

**8. 3x² – 7x + 6 < 0**

(a) 0.66 < x< 3

(b) -0.66 < x< 3

(c) -1< x< 3

(d) None of these

**9. |x² – 2x|< x**

(a)1<x<3

(b)-1<x<3

(c) 0 < x< 4

(d) x > 3

**10. Kamla got married 6 yr ago. Today, her age is 1 1/4 times of her age at the time of marriage, Her son’s age is 1/10 times her age. Her son’s age is**

(a) 4 yr

(b) 5 yr

(c) 2 yr

(d) 3 yr

**11. A father is 30 times older than his son. 18 yr later, he will be only thrice as old as his son. Father’s present age (in yr) is**

(a) 25

(b) 30

(c) 40

(d) 45

**12. 3 yr ago x’s age was double of y’s seven years,hence the sum of their united ages Will be 83 yr. The age of x today is**

(a) 47 yr

(b) 35 yr

(c) 45 yr

(d) 24 yr

**13. |x² – 3x|+x-2 < 0**

(a) (1 – √3) < x< (2 + √2)

(b) 0 < x < 5

(c) (1 – √3, 2 – √2)

(d) 1 < x < 4

**14. x² – |5x – 3 | – x < 2**

(a)x > 3 + 2√2

(b)x < 3 + 2√2

(c)x > – 5

(d) -5 < x < 3 + 2√2

**15.|x – 6| > x²**** – 5x+9**

(a)1≤x<3

(b)1<x<3

(c)2<x<5

(d)-3<x<1

**16. Eight consecutive numbers are given. If the average of the two numbers that appear in the middle is 6, then the sum of the eight given numbers is SSC (10 + 2) 2012**

(a) 36

(b) 48

(c) 54

(d) 64

**17. If the sum of five consecutive integers is S, then the largest of those integers in terms of S is SSC (10 + 2) 2011**

(a)S – 10/5

(b)S + 4/4

(c)S + 5/4

(d)S + 10/5

**18. A drum of kerosene is 3/4 full. When 30 L of kerosene is drawn from it, it remains 7/12 full. The capacity of the drum is SSC (10 + 2)2010**

(a) 120 L

(b) 135 L

(c) 150 L

(d) 180 L

**Answers**

**Hints & Solutions**

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