August 11, 2010

Quantitative Ability - Linear Equations At A Glance

A variable is a symbol that represents a number. Usually we use lower case letters such as x, y, n, t, etc for variables. For example, we might say that s stands for the side-length of a square. We now treat s as if it were a number we could use. The perimeter of the square is given by 4s. The area of the square is given by s2.

When working with variables, it can be helpful to use a letter that will remind you of what the variable stands for:
Let n be the number of people in a movie theater .
Let t be the time it takes to travel somewhere.
Let d be the distance from my house to the park.

Variables can be operated upon in a manner similar to real numbers. Thus you can say that the H.C.F of x2, x3 and x5 is x2 and that the reciprocal of 2y is 1/2y.
• Remember that strictly speaking, H.C.F of x2, x3 and x5 is x2 only when x is a natural number or at best a positive real number. By extending the definition of H.C.F to negative numbers, we may go wrong. For example, it is wrong to say -16 is the L.C.M of – 8, - 4 and – 2 because – 16 is a common multiple and is smaller than – 8.

Expressions
An expression is a mathematical statement that is a combination of numbers, variables, or both.

The following are examples of expressions:
2x + 7
2xy+5
2+6x (4-2)
z/3 x (8-z)

Example.- Anil weighs 70 kilograms, and Anita weighs k kilograms. What is the expression that gives their combined weight?
Sol: The combined weight in kilograms of these two people is the sum of their weights, which is 70 + k.
Care should be taken while putting together numbers and variables since they represent physical values.
While adding or subtracting them, we need to check that they have the same units.
If in the above problem, Anita’s weight was given in grams then the expression for the weight in kilograms would be 70+ k/1000. We could also express the combined weight in grams as 70 x 1000 + k.
To evaluate an expression at some number means replace a variable in an expression with the number, and simplify the expression.

E.g.:
To evaluate the expression (4 z + 12) when z = 15, we replace each occurrence of z with the number 15, and simplify using the usual BODMAS rules.

4z + 12 becomes 4 × 15 + 12 = 60 + 12 = 72

Evaluation can be done for expressions with multiply variables as well. In this case the values of each of these variables should be given for which the expression is to be evaluated.

E.g.: To evaluate x + 3y + xz when x = 1, y =2 and z =3, we replace the variables by their values to get 1 + 3 × 2 + 1 × 3 = 10.

An expression having only one term is called a monomial. So x, v, 2xyz, ab, 9abx2y, …etc. are all monomials. The constant that appears in any term other than the algebraic variable is called the ‘coefficient’ (1,2, 9, …, in the examples given).
An expression having 2 terms is called binomials, so ab+abc is a binomial.
An expression having 2 or more than two terms e.g. X + Y + 5Z, 2X2 + 4YZ3 + 3XZ2, …, etc. is called polynomials. The point to note is that the terms are separated by an addition or subtraction sign.
So, how we differentiate between x, 2xyz and 9abx2y?? These all are Monomials.
We define degree of an algebraic expression. The DEGREE of an expression is equal to the maximum values of the sum of the powers of the variables in any term. You should note two things in the definition, sum of powers and maximum sum in any term. In the above examples of monomials, x has 1 degree, 2xyz is 1+1+1=3 degrees (1 each for x,y,z) and 9abx2y has 5 degrees (1each for a,b,y and 2 for x).
Maximum sum in any term applies for binomials or polynomials. So for Polynomial, 2X2 + 4YZ3 + 3XZ2, we have three different terms. Term 1 has a degree of 2, term 2 has a degree of 4 and third term has a degree of 3. As the maximum value is 4, we say that the degree of polynomial is 4.

Expressions of Degree 1 are called LINEAR expressions, of Degree 2 are called QUADRATIC expressions and of Degree 3 are called CUBIC expressions. (There are names for expressions with degree 4 ,5 and so on but let us leave them for mathematicians)

3. Equations
An equation is simply a statement that two numbers or expressions are equal. Equations are useful for relating variables and numbers. Many word problems can easily be written down as equations. There are some simple rules for simplifying equations.
The following are examples of equations:
2 = 2
x = 7
7 = x
t + 3 = 8
w + 4 = 12 – w
3 Í (d + 4) – 11 = 321 – 23.

Example: 20 added to my age in years, y, is equal to four times my age, minus 10.

Sol: The first expression stands for “20 added to my age in years”, which is y + 20
This is equal to the second expression for “four times my age, minus 10”, which is 4 y – 10
Setting these two expressions equal to one another gives us the equation y + 20 = 4 y – 10.

Example: When 17 is subtracted from 5 times a certain number, the result is 73. What is the number? 
Sol: 5x – 17 = 73,
x = 18

Solution of an Equation
When an equation has a variable, the solution to the equation is the value of the variable that makes the equation true.

E.g.: We say y = 3 is a solution to the equation 4* y + 7 = 19, for replacing each occurrence of y with 3 gives us
4 × 3 + 7 = 19
So, 12 + 7 = 19 so, 19 = 19, which is true.
Y = 10 is NOT a solution to the equation 4 × y + 7 = 19. When we replace each y with 10, we get
4 × 10 + 7 = 19
So 40 + 7 = 19 or, 47 = 19, which is not true.

• It is not necessary that every equation should have a solution. For instance x = x + 3 has no solution.
An equation can have multiple solutions, the number of solutions being finite or infinite. z2 = 4 has both z = 2 and z = -2 solutions. 0/y = 0 has infinite solutions for y since the equation is true for any real value of y other than 0.
• If we have equations with multiple variables, the solution consists of values for each of the variables for which the equation is true.
A solution of the equation x + 2y = 3 is x = 1 and y = 1; x = 2 and y = ½ is another. As you can see, there are infinite solutions for the equation.
Simplifying Equations

To find the solution for an equation, we can use the basic rules of simplifying equations. These are as follows:
• You may evaluate any parentheses, exponents, multiplications, divisions, additions, and subtractions in the usual order of operations (BODMAS).
• You may combine like terms. This means adding or subtracting variables of the same kind. The expression 2x + 4x simplifies to 6x. The expression 13 – 7 + 3 simplifies to 9.
• You may add any value to both sides of the equation.
• You may subtract any value from both sides of the equation. This is equivalent to adding a negative value to each side of the equation.
• You may multiply both sides of the equation by any real number.
• You may divide both sides of the equation by any number except 0.

Example. To simplify an equation like 2x – 12 + 20 = 36, we do the following in accordance with the above rules.
Sol: 2x + (-12) + 20 = 36
2x + 8 = 36
Add the two like terms (the integers)
2x + 8 - 8 = 36-8
Subtract 8 from each side of the equation, (or add a –8 to each side).
So. 2x = 28
2.x. ½ = 28 x ½ Divide both sides by 2 (or multiply by ½)
Or, x = 14

• Remember that dividing both sides by zero is not allowed. We commonly strike off multiplicative terms from both sides (“cancelling”).
Such cancelling is not allowed if the term being struck off is zero. Thus, if you are cancelling a variable you need to be sure that it cannot take the value of 0.
To illustrate, let’s take x = 2.
This can also be written as: 3x – 2x = 6 – 4
Rearranging the terms, 2x – 4 = 3x – 6 or, 2 (x – 2) = 3 (x – 2)
Cancelling (x – 2), we get 2 = 3.
• The flaw obviously is that when we are cancelling x- 2, we are actually performing a division by x - 2.
But since x = 2, x - 2 is zero and so we cannot divide by it.

Example.
A car travels down the freeway at 55 kilometers per hour. Write an expression for the distance the car will have travelled after h hours?
Sol:
Distance equals speed time’s time, so the distance travelled is equal to 55 h.

Example. There are 2000 litters of water in a swimming pool. Water is filling the pool at the rate of 100 liters per minute.
Write an expression for the amount of water, in liters, in the swimming pool after m minutes?
Sol:
The amount of water added to the pool after m minutes will be 100 liters per minute time m, or 100 m.
Since we started with 2000 liters of water in the pool, we add this to the amount of water added to the pool to get the expression 100 m + 2000.

Example. Evaluate the expression (1 + z) × 2 + 12/3 - z when z = 4?
Sol: We replace each occurrence of z with the number 4, and simplify using the usual rules: parentheses first, then exponents, multiplication and division, then addition and subtraction.
(1 + z) 2 + 12/3 - z becomes (1 + 4) 2 + 12/3 - 4 = 5 × 2 + 12/3 - 4 = 10 + 4 – 4 = 10.
After understanding the basics of equations, let us understand Linear equations. Linear equations are equations with one degree.

If we look at the different solutions, they may appear in three forms.

1. UNIQUE: There is only one set of values for the variables that satisfy the equation.
2. INFINITE: The number of sets of values of the variable that satisfy the equation is infinite.
3. NO SOLUTION: There is no set of values of the variables that satisfy the given equation.

For instance, take the case of the equation we have considered X + Y =4
If we try to solve this equation, we would find that the following values of X and Y would satisfy the equation.
(X, Y) = (1, 3) or (1.1, 2.9) or (2.5, 1.5) or … so on. If we try to list down all the possible values, possible set, there are infinite possibilities. So the number of solutions for (X, Y) is infinite.
Linear equation in 1-variable
These are the equations having only one unknown variable.
General form is:
Ax + B = 0
Where A and B are constants.
For finding unique value of variable ‘x’ we need only one equation.
Linear equation in 2-variable
These are the equations having two unknown variables.
General form is:
Ax + By + C = 0
Where A, B and C are constants.
For finding unique value of variables ‘x’ and ‘y’ we need two independent and consistent equations.
Linear equations in 3-variables.
These are the equations having three unknown variable.
General form is:
Ax + By + Cz + D = 0
Where A, B, C and D are constants.
For finding unique value of variables ‘x’, ‘y’ and ‘z’ we need three independent and consistent equations.
Linear combination of equations or Dependent equations
Lets suppose there are ‘n’ number of linear equations: l1, l2, l3 ……ln
And there are ‘n’ constants k1, k2, k3 ……….. kn
Then these equations will be said to be in linear combination if:
K1 l1 + k2 l2 + k3 l3 ……. Kn-1 ln-1 = kn ln
So we can say that, if any equation can be written as the linear combination of some other equations then these equations are dependent on each other.
Graphical Approach
If all these points are represented as coordinates on a graph and joined together, they form a straight line. The graph would look as below.


On the contrary, an equation or a system of equations having a UNIQUE solution has only one set of the variables that satisfy the equation.
An equation or a system of equations has a unique solution if and only if the number of variable is equal to or less than the number of independent and consistent equations.

Slightly confused!!!! Let us break the above definition to make more sense.
Independent
An equation is not independent if it is derived as a combination of any of the other equations in the system of equations.

For example, suppose there are three equations.

X + Y = 4 …(i)
2X + 3Y = 5 …(ii)
4X + 5Y = 13 …(iii)
The equation (iii) is not independent of the other two since it can be formed by
2(i) + (ii)
But in this case as we have only two variables, we may still get Unique solution for X and Y.



To be Contd.


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