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An equation whose solution is a straight line. In a linear equation, the variables are raised to the first powerâ€”there are no variables in denominators, no variables to any power (other than one), and no variables under root signs.
For Example 2x + 4 = 0 --- eq(1)
Solving linear equations means finding out the unknown (usually only one but possibly several). In the above equation x is the unknown but there can be more than 2 or more variables (unknown) in a linear equations as given below.
2x + 3y = 12 --- eq(2)
6x +8y +9z =12 --- eq(3)
So, A linear equation is an equation that can be written in the form
y= a x + b
where x and y are variables & a and b are constants
Note that the exponent on the variable of a linear equation is always 1.
These are examples of linear expressions:
x + 4
2x + 4
2x + 4y
Following equations are not linear expressions: x2 (no exponents on variables)
2xy + 4 (can't multiply two variables)
2x / 4y (can't divide two variables)
âˆšx (no square root sign on variables)
A value, such that, when you replace the variable with it, it makes the equation true.
In equation 1 if we put x=-2 then it satisfies the condition i.e 2(-2)+4=0, So 2 is the solution for eq(1)
Like so the values of (x,y) that satisfies eq(2) are (3,2)Now letâ€™s solve one equation step by step
Example 1: Find the for y: 7y + 5 - 3y + 1 = 2y + 2
First combine the similar terms on the left side.
We'll start with 7y and -3y.
(Don't forget to take the sign in front of the term. If there isn't a sign in front of the term, it is considered +.)
7y - 3y = 4y. So we have: 4y+6=2y+2
=> 4y â€“ 2y = 2 -6
=> 2y = -4 or y = -2
Linear inequalities are solved much the same way as linear equations with one exception: when multiplying or dividing both sides of an inequality by a negative number the inequality sign must be reversed For example 2 < 3 but -2 > - 3. Adding and subtracting the same quantity to both sides of an inequality never changes the direction of the inequality sign.
For CAT you should be solving these equations in seconds with out requiring pen and paper
1. x - 4 = 10 ans 14
2. 2x - 4 = 10 ans 7
3. 5x - 6 = 3x - 8 ans 7
4. 2(3x - 7) + 4 (3 x + 2) = 6 (5 x + 9 ) + 3
5. 2x + 8 â‰¥ 4 ans x â‰¥-2
6. 10â‰¥ 2x + 6 â‰¥ 4
7. If 0.16x + 1.1 = 0.2x + 0.95, then x =? Ans
8. If A = Â½(2p+c), then c=?
9. 4(x-5) â€“ 3(6-2x) = 2 ans
Now letâ€™s move to linear equation involving more than one variable , for ex.
3x + 2y = 12 and 3x + 5y = 18
Above two equations are quite easy to solve, one of the variable x has same co-efficient in both the lines, so we can easily make it as 5y-2y= 18- 12 or 3y=6 or y=2 and thus x=8/3
Now if there is no variable with same co-efficient then we have to make it by multiplying with a factor.
4x + 7y = 15 -------eq(4)
2x + 3y = 11 -------eq(5)
Above two equations can be written as
8x + 14y = 30 -------eq(6)
8x + 12y = 44 -------eq(7) We have multiplied eq(4) with 2 and eq(5) with 4
Now eq(6) and (7) can be easily solved but the big question is how to know what to multiply? (left for you to figure out).
In CAT you cant expect to get any direct questions from this section but If you practice a lot in solving linear equations with out using pen & paper then it will help in solving other questions quickly.
Few Questions are given below and please do ask for help if you require any.
1. Find two consecutive numbers whose sum is 57.
2. Find three consecutive numbers whose sum is 48.
3. Find four consecutive numbers whose sum is 90.
4. The sum of two numbers is 85. One number is 15 more than the other. What are
the two numbers?
5. The sum of two numbers is 48. One number is three times the other. What are the numbers?
6. A cable TV company charges Rs.21.95 per month for basic service. Each premium channel selected costs an additional Rs. 5.95 per month. If x represents the number of premium channels selected, which expression can be used to find the monthly cost of cable service?
7. Steve ran a 12-mile race at an average speed of 8 miles per hour. If Adam ran the same race at an average speed of 6 miles per hour, how many minutes longer than Steve did Adam take to complete the race?
8. If Nathan is Â¼ as old as his father and the sum of their ages is 60, then how old is Nathan?
9. How much 10% alcohol solution should be mixed with 14 ounces of 18% solution to get a 12% solution?
10. The perimeter of a rectangle is 56 inches and the width is three fourths of the length. What is the length?
11. A piggybank contains Rs. 3.20. There are twice as many nickels as quarters and half as many dimes as quarters. How many dimes are in the piggybank?
12. A car and passenger train pass each other at noon. The train is eastbound and its average speed is 45 mph. The car is westbound and its average speed is 60 mph. When will the car and train be 14 miles apart?
13. A rectangular piece of cardboard starts out with its width being three-fourths its length. Four inches are cut off its length and two inches from its width. The area of the cardboard is 72 square inches smaller than before it was trimmed What was its original length and width?
14. A rectangleâ€™s length is one-and-a-half times its width. The length is increased by 4 inches and its width by 3 inches. The resulting area is 97 square inches more than the original rectangle. What were the original dimensions?
15. A boxâ€™s width is 2/3 its length. The perimeter of the box is 40 inches. What are the boxâ€™s length and width?
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