linear equation lecture

3.  A pair of linear equations in two variables is
said to form a system of simultaneous linear
equations.
 For example: 2x – 3y +4 = 0 and x + 7y -1 = 0
 These two equations form a system of two
linear equations in variables x and y.

4.  The general form of a linear equation x and y is
 Ax + by +c = 0, where a and b is not equal to zero
and are real numbers.
 A solution of such an equation is a pair of
values. One is for x and the other for y. Once
the values of x and y are represented the two
sides of the equation hold for them to be equal.
 Every linear equation in two variables has
infinitely many solutions which can be
represented on a certain line.

5.  Let us consider the following system of two
simultaneous linear equations in two variable:
 2x-y=-1 and 3x+2y=9
 Here we assign any value to one of the two
variables and then determine the value of the
other variable from the given solution by using
a t-chart

6. For the equation:
 2x-y=-1 solve for y
y=2x+1 Plug x=0,2 to get y values
 3x+2y=9 solve for y
y=9-3x Plug x=3,-1 to get y values
Any questions on how to
Complete t-chart?
x 0 2
y 1 5
x 3 -1
y 0 6

7. X Y
0 1
2 5
X Y
3 0
-1 6
Y=2x+1
Y=9-3x
0
1
2
3
4
5
6
7
-2 -1 0 1 2 3 4
Y-Values

8.  The most common used algebraic methods of
solving simultaneous linear equations in two
variables are:
 Method of by substitution.
 Method by equating the coefficient.
 Method by elimination

9.  Solve the equations given by solving for x and
substitute the x-value of (i) first equation into (ii)
second equation to get one equation
(i) x+2y=-1 and (ii)2x-3y=12
(i): x=-2y-1 plug this equation into (ii) by substituting value of x
2(-2y-1)-3y=12 distribute and simplify
-4y-2-3y=12 combine like terms and solve for y
-7y-2=12
-7y=14
y=-2 putting the value of y in equation (i) w get
x=-2y-1
x=-2(-2)-1
x=3 Hence solution of the equation is (3,-2)

10.  Try on your own
 Use previous slide to solve for x and y in these
two equations given:
 y=5x-1 and 2y=3x+12

11.  y=5x-1 (i) and 2y=3x+12 (ii)
Step 1: substitute (i) into (ii) from y value.
Step 2: 2(5x-1)=3x+12 distribute
Step 3: 10x-2=3x+12 combine like terms to solve for x
Step 4: 7x-2=12
7x=14
x=2
Step 5: Use value of x to solve for y in (i) equation
Step 6: y=5(2)-1 y=9
Step 7: Solution: x=2 and y=9, check your work by
plugging in these values in both equations to make
sure left equals right in both equations.
Any questions?

12.  In this method, we eliminate one of the two
variable which can easily be solved.
 Putting the value of this variable in any of the
given equations. The value of the other variable
can be obtained.
 For example we want to solve:
 3x+2y=11 and 2x+3y=4

13.  (i) 3x+2y=11 and (ii) 2x+3y=4
 First we can use a method by equating the coefficient
which means multiplying (i) by 3 and (ii) by -2.
 We get 3(3x+2y=11)------9x+6y=33
-2(2x+3y=4)------- -4x-6y=-8
Now we can eliminate y values by adding both (i) and (ii)
9x+6y=33
+ -4x-6y=-8
----------------
5x=25 solve for x
x=5

14.  Putting the value of y in equation (i)
 3x+2y=11
 3(5)+2y=11
 15+2y=11
 2y=-4
 y=-2
 Solution: x=5 and y=-2
 Check solutions

15.  Try on your own
 Use previous slides to solve for x and y in these
two equations given:
 x+3y=-5 and 4x-y=6

16.  x+3y=-5 (i) and 4x-y=6 (ii)
 Step 1: Multiply (i) by -4 to eliminate x.
 Step 2: -4(x+3y=-5)------- -4x-12y=20
 Step 3: Add both equations and combine like terms to
eliminate x.
 Step 4: 4x-y=6
+ -4x-12y=20
----------------------
-13y=26
-y=2…..multiply by (-1)….. y=-2
Step 5: Plug in y=-2 into equation (i)
x+3(-2)=-5…. x-6=-5 x=1
Step 6: Check solutions

17.  Given two equations, with two variables, we
are then able to use different methods to solve
for x and y
 Once we found the values of x and y, are we
finished?
 We need to make sure both values hold for both
equations.
 What are the two methods we went over?
 The methods are methods by substitution and
method by elimination.
 Any questions?