Solved Examples: Linear Equations | Quantitative Aptitude for SSC CGL PDF Download

Definition of Linear Equation

A linear equation is a mathematical expression that depicts a straight line within a two-dimensional Cartesian coordinate system. This equation is commonly expressed in the form y=mx+b.

Tips to solve Linear Equations

Here are some useful tips to help you solve linear equations efficiently:

• Isolate the Variable: The objective is to relocate the variable (typically denoted as 'x') to one side of the equation. Employ inverse operations (addition, subtraction, multiplication, division) to shift all other terms to the opposite side.
• Combine Similar Terms: If there are multiple terms with the variable on the same side, amalgamate them to simplify the equation.
• Utilize the Distributive Property: When faced with parentheses, distribute the terms within them to eliminate the parentheses.
• Eliminate Fractions: If fractions are present in the equation, multiply both sides by the common denominator to remove them.
• Maintain Equation Balance: Ensure equality by performing the same operation on both sides of the equation.
• Simplify Radicals: If square roots or other radicals are present, attempt to simplify them by identifying perfect square factors.
• Be Mindful of Negative Numbers: Pay attention to negative signs and avoid sign errors during calculations.
• Verify Your Solution: After determining the variable's value, substitute it back into the original equation to confirm its satisfaction.
• Regular Practice: Like any skill, regular practice is essential for mastering linear equations. Solve various problems to enhance proficiency.
• Learn from Common Mistakes: Recognize common mistakes made by students when solving linear equations and steer clear of them.
• Graphical Representation: At times, graphing the equation on a coordinate plane aids in visualizing the solution.
• Tackle Word Problems: Transform word problems into linear equations by defining variables and setting up the equation before proceeding with the solution.

Rules for Linear Equations Questions and Answers

• If a=b then a+c=b+c for any c. All this is saying is that we can add a number, c, to both sides of the equation and not change the equation.
• If a=b then a−c=b−c for any c. As with the last property we can subtract a number, c, from both sides of an equation.
• If a=b then ac=bc for any c. Like addition and subtraction, we can multiply both sides of an equation by a number, c, without changing the equation.
• If a=b then a/c=b/c for any non-zero c. We can divide both sides of an equation by a non-zero number, c, without changing the equation.

Examples

Example 1: The roots of the equation 4x − 3 × 2x + 2 + 32 = 0 would include-
(a) 20
(b) 40
(c) 5
(d) 17
Ans: (d)
4x−3×2x+2+32=0
4x−6x+34=0
−2x+34=0
2x−34=0
2x=34
x=17

Example 2: If x = 1+21/2 and y=1-21/2, then x² + y² is -
(a) 107/2
(b) 203/2
(c) 997/2
(d) 445/2
Ans: (d)
x = 1+21/2 and y=1-21/2
x²+ y² 



2 + 441/2
445/2

Example 3: 2x+y=2 and  2x - y = 21/2 ,the value of x is:
(a) 10/14
(b) 9/8
(c) 25/8
(d) 5/4
Ans: (c)

2x + y = 2
2x - y = 21/2
Adding both equations we get:

4x = 25/2
x = 25/8
Putting this in

Example 4: What is the slope of the line represented by the equation 3y+4x=12?
(a) 3/4
(b) −4/3
(c) 3
(d) −4
Ans: (b)
To find the slope of the line, we need to rewrite the equation in slope-intercept form (y=mx+b), where m is the slope.
Solving for y gives
The coefficient of x represents the slope, so the correct answer is −4/3

Example 5:  If 6(x-3) = 36(x-5), then what is the value of x?
(a) 51/5
(b) 27/5
(c) 15/7
(d) 17/3
Ans: (b)
6(x-3) = 36(x-5)
x - 3 = 6(x - 5)
x - 3 = 6x - 30
5x = 27
x = 27/5

Example 6: If ax = b, by = c and cz = a, then the value of xyz is:
(a) 4
(b) 3
(c) 2
(d) 1
Ans: (d)
ax = b, by = c and cz = a
x = b/a
 y = c/b
 z = a/c

xyz = 1

Example 7: If 4x + 3 = 2x + 7, then the value of x is:
(a) 2
(b) 3
(c) 1
(d) 4
Ans: (a)
4x+3 = 2x+7
2x = 4
x = 2

Example 8: If x = 8, y = 27, the value of
(a)
(b)
(c)
(d)
Ans: (b)

Example 9: If 2x + 3y = 16 and 2x - 3y= 36, the value of x is:
(a) 13
(b) 23
(c) 33
(d) 43
Ans: (a)
2x + 3y = 16 -----(1)
2x - 3y= 36 ------ (2)
adding both eq we get
4x = 52
x = 13

Example 10: Determine whether the ordered triple (3,−2,1) is a solution to the system.
2x+y+z=5,
6x−4y+5z=31,
5x+2y+2z=13
(a) True
(b) False
(c) No solution
(d) None of the above
Ans: (a)
We will check each equation by substituting the values of the ordered triple for x,y, and z
x+y+z=2(3)+(−2)+(1)=5 True
6x−4y+5z=6(3)−4(−2)+5(1)=18+8+5=31 True 5x+2y+2z=5(3)+2(−2)+2(1)=15−4+2=13 True
The ordered triple (3,−2,1) is indeed a solution to the system.