A linear equation is a mathematical expression that corresponds to a straight line within a twodimensional Cartesian coordinate system. It takes the form y=mx+b, where x and y represent variables, m denotes the slope of the line, and b signifies the yintercept.
There are several shortcuts and techniques to solve linear equations quickly and efficiently. Here are some useful shortcuts:
Employing these quick methods and strategies not only saves time but also enhances your proficiency in solving linear equations. Nevertheless, it's crucial to retain an understanding of the principles behind each shortcut to maintain accuracy in your solutions.
Example 1: If 3a + 6 = 4a − 2, then find the value of a?
(a) 3
(b) 8
(c) 6
(d) 7
Ans: (b)
We can use the trick of eliminating the option
Option 1, put a = 3
3 * 3 + 6 = 15
4 * 3 2 = 10
This means option 1 is incorrect.
Now, check for option 2, put a = 8
3 * 8 + 6 = 30
4 * 8 – 2 = 30
This means option 2 satisfies the equation. Therefore, it is the correct option.
Example 2: Which of the following is the correct equation of the line passing through the points (2, 5) and (4, 11)?
(a) y = 3x + 1
(b) y = 2x + 1
(c) y = 2x + 3
(d) y = 3x + 5
Ans: (a)
To find the equation of a line passing through two given points, we first calculate the slope (m) using the formula:
For the points (2, 5) and (4, 11), the slope is
Next, we use the pointslope form of the linear equation:
y – y_{1} = m(x – x_{1})
Plugging in the values (2, 5) and m = 3, we get the equation y – 5 = 3(x – 2). Solving for y, we find y = 3x + 1.
Example 1: The cost of 5 blankets and 6 bedsheets is Rs.1500. The cost of 6 blankets and 5 bedsheets is Rs.1300. Find out the total cost of one blanket and one bedsheet.
(a) Rs. 255
(b) Rs. 250
(c) Rs. 81.81
(d) Rs. 254.545
Ans: (d)
Let the cost of blankets be x and the cost of bedsheets be y.
According to the question:
5x+ 6y= 1500…(1)
6x+ 5y=1300…(2)
Multiply Eq 1 by 5 and Eq 2 by 6,
we get.
25x+30y = 7500…(3)
36x+30y = 7800…(4)
Subtract equation (3) from equation (4)
11x = 300
x = 300/11
y = 2500/11
Total cost = x+y
= 2800/11 = 254.545
Example 2: What is the xintercept of the line represented by the equation 2x + 3y = 12?
(a) 4
(b) 6
(c) 8
(d) 12
Ans: (b)
To find the xintercept, we set y = 0 and solve for x.
Substituting y = 0 into the equation 2x + 3y = 12, we get 2x + 3(0) = 12. Simplifying, we find 2x = 12, and then x = 6.
Example 3: If the line 3x – y = 5 is parallel to the line 2x + ky = 8, what is the value of k?
(a) 3
(b) 2
(c) 2
(d) 3
Ans: (b)
Two lines are parallel if their slopes are equal.
The slope of the line 3x – y = 5 can be found by rearranging the equation in slopeintercept form (y = mx + b), where m is the slope.
So, 3x – y = 5 becomes y = 3x – 5, and the slope is 3.
The line 2x + ky = 8 can also be rearranged as y = (2/k)x + 8/k, where the slope is 2/k. To make both slopes equal (3 and 2/k), we need 2/k = 3. Solving for k, we find k = 2/3.
314 videos170 docs185 tests

314 videos170 docs185 tests


Explore Courses for SSC CGL exam
