LINEAR EQUATIONS
Equation : It is defined as a statement of equality which involves one or more variables.
Linear equation : If an equation has only linear polynomial, it is called a linear equation.
Solution of a equation : It is the value of the variable which when substituted in the equation, makes its two sides equal.
Solving a linear equation :
- Same number can be added to both sides of the
- Same number can be subtracted from both sides of the
- Both sides of the equation can be multiplied by the same
- Both sides of the equation can be divided by the same non-zero
If lx + b =
p then
mx + d q
q (lx + b) = P(mx + d) is called the cross multiplication.
Uses of linear equations : Three steps are used to solve problems based on linear equations :
- Express the unknown quantity by
- From the information, make a linear equation in x.
- Solve the linear equation to get the value of
RULES FOR SOLVING LINEAR EQUATIONS IN TWO VARIABLES
Linear equations of two variables : An equation having the form ax + by + c = 0, where a, b, c Î R, a ¹ 0, b ¹ 0, and x, y are two unknowns is called linear equation of two variables.
Solution : The values of x and y which satisfies the equation ax + by + c = 0 where a ¹ 0, b ¹ 0, a, b, c Î R is known as the solution of the equation.
Method of Substitution : 1. From the two given equations, express one of the two unknowns in terms of the other.
- Put the value of unknown into the other equation. By solving this equation, we get the value of one
- Put the value of this unknown in either of the two equations. By solving that, we get the value of the other unknown.
Eliminator method : (a) Multiply both the equation by such numbers which make the coefficient of one of the two unknown numerically equal.
- After this add or subtract so as to get an equation containing only the other By solving these equations, the value of the one unknown is calculated.
- Put the value of this unknown in either of the By solving this, the value of the other unknown will be obtained.
Condition for different kinds of solution of linear equations : We consider the system of two linear equation in two variables :
a1x + b1y + c1 = 0 a2x + b2y + c2 = 0
Case 1. When (a1b2 – a2b1) ¹ 0 i.e.,
a1 ¹
b1given by x =
b1c2 – b2c1
a2 b2 a1b2 – a2b1
and y =
c1a2 – c2a1 a1b2 – a2b1
This usually we write as
x = y = l
b1c2 – b2c1 c1a2 – c2a1 a1b2 – a2b1
The above solution can be remembered by the following diagram x y l
b1 c1 c1 a1 a1 b1
b2 c2 c2 a2 a2 b2
Numbers with downward arrow are multiplied first and from their product, the
product of numbers with upward arrow is to be subtracted.
Case 2. When
a1 =
b1 =
c1 = k, then the equation have an
a2 b2 c2 infinite number of solutions.
Case 3. When
a1 =
b1 ¹ c1
of y.
a2 b2 c2
Clearly, in this case the equations have no solution i.e., it is inconsistent.
How to draw the graph of a linear equation : 1. Rewrite the equation in terms
- Draw the value table for x and y by taking any three convenient values for x,
find the corresponding value of y.
- Draw the two axes on a square paper and mark the
- Plot three ordered pairs from the
- Join the three points by using a straight scale and produce on either to run completely across the
- While making a table for the values for x and y, values should be taken both positive and negative values. It should be ensured that the points that are plotted are not too
- Simultaneous linear equations in two variables : When two or more equa- tions are satisfied by the same set of values of two variables, then these are called as simultaneous equations. The pair of values as x and y satisfying each one of the given equations is called a solution of the
- Consistent and inconsistent system of simultaneous linear equations : A system having two simultaneously linear equations are said to be consistent, if they have at least one solution. Otherwise, if there does not exist any solution of the system it is said to be
- Rules of solving simultaneous linear equations : Three methods for solv- ing simultaneous equations are They are as follows :
q Method of substitution,
q Method of elimination,
q Method of cross-multiplication.
Note : If in ax + by + c = 0, we have a = 0, b ¹ 0 or b = 0, a ¹ 0 then equation reduces to by + c = 0 or ax + c = 0 respectively. In either case we get an equation in one variable. That is why both a and b are assumed to be non-zero.
- Grapical method of solving simultaneous equations : Suppose two lines l and m represent the graphs of two equations respectively. The following three possibili- ties arise :
- Exactly one solution, if l and m are intersecting
- Infinitely many solutions, if l and m are coincident; and
- No solution, if l and m are parallel
To solve a problem, we follow three steps : Make assumptions, using two variables, say, x and y.
From the information given formulae two equations in terms of x and y. Solve these equations simultaneously and varify the results.
Properties of equality :
- Add the same quantities to both sides of an equality without changing
- We can subtract the same quantity from both sides of an equality without changing
- Multiply both sides of an equalty by the same number without changing
- Divide both sides of an equality by the same non-zero number without changing
