# Lesson Notes By Weeks and Term - Junior Secondary School 3

SIMULTANEOUS EQUATION

SUBJECT: MATHEMATICS

CLASS:  JSS 3

DATE:

TERM: 2nd TERM

WEEK ONE

TOPIC: SIMULTANEOUS EQUATION

Equation such as 4x + 1=7, has only one solution and one unknown, thus it is called linear equation. Considering equations such as 4x +2y = 24 which contains two unknown quantities (x,y) it cannot be solved unless one of the variables is given or another connecting the variable is given.  Hence we have two linear such as x+y=10; and x-y=2, this is known as simultaneous equation. To solve the given equations (simultaneous equation), we need to find the value of x and value of y that will satisfy both equations at the same time.

SUBSTITUTION METHOD OF SOLVING SIMULTANEOUS EQUATION

In using this method, one of the variables is made the subject of the equation. Then the value of the subject of the equation is substituted in the second equation. When the substitution is done, the equation is solved to obtain the value of one of the variables. The value is then substituted in one of the pair of equations to find the second variable.

Example: solve this pair of simultaneous equation using substitution method;

X+6y = -2 ; 3x+2y =10

Solution:

X+6y = -2………(1)

3x+2y=10 ………(2)

From eq (1) x= -2 -6y

Sub x= -2 -6y in eq(2)

3(-2-6y) +2y = 10

-6-18y +2y =10

-16y =10+6

-16y =16

y=16/-16

y=-1

sub y = -1 in eq(1)

x +6y= -2

x +6(-1) = -2

x-6 = -2

x=-2+6

x = 4

Evalution:

1. x+3y =-4 ;  x +y=-10           2. 5x-y=35; 3x-2y=14

ELIMINATION METHOD OF SOLVING SIMULTANEOUS EQUATION

In the Elimination method, the two simultaneous equations are either added or subtracted so as to eliminate one of the variables. This is very useful to solve simultaneous equation especially when none of the coefficient of the unknown is one (1).

Example: use Elimination method to solve; 6x +5y=2 and x-5y=12

Solution:

6x+5y=2

X-5y=12

x=2

sub x=2 in eq(1)

6x+5y=2

6(2)+5y=2

12+5y=2

5y= 2-12 ;         5y=-10

y= -10/5 ;     y= -2

EVALUATION:  simplify using Elimination method;

1. 2y-x=10; y+x=2      2. 4p+3q ;   3p-5q=-10

WORD PROBLEMS LEADING TO SIMULTANEOUS EQUATION

To solve such problems:

1. Identify the two unknowns and represent them with letters.
2. Translate the words into equations.
3. Use any convenient method to solve the two unknowns.

Example:

The difference between the ages of Audu and Ojo is 15. if the sum of their ages is 47. How old are they?

Solution: let x represents Audu’s age and y represent Ojo’s age.

x-y =15…………(1)

x+y =47………..(2)

x= 62/2   =31

Substituting:        x-y=15

31-y=15

-y=15-31

-y=-16

y=16

EVALUATION

1. The cost of one orange and two apples is 24k. Two orange and three apples cost 44k. How much does each cost?
2. Six pencils and three rubbers cost N117. Five pencils and two rubbers cost N92. How much does each cost?

Essential mathematics by A.J.S OLUWASANMI Pg 148-152

Exam focus for J.S.S CE Pg 220-222

WEEKEND ASSIGNMENT

1. If X=2, the value of y in y=8-4x is A. 1 B. 2 C.3
2. If the equation y = mx+c is satisfied by x=1, y=5 and c=0, the value of m is A. 3 B. 5 C. 2
3. the solution of x if y=5x+2; and x+2y=15 is A. 1 B. 2 C. 5
4. The sum of two numbers is 18 and their difference is 12. Find the two numbers from the above question.A.6&10 B.15&3 C. 10&3
5. What is the product of the two numbers A. 60 B.45 C. 30

THEORY

1. The sum of the ages of a man and his wife is 73yrs. Eight years ago the husband was twice as old as the wife. How old are they now?
2. The below is an equilateral triangle triangle with the dimensions shown:

2p                            5q-2

p+q+5

Find

1. The value of p and q
2. The perimeter of the triangle in meters
3. The area of the triangle to 3.s.f.g

WEEK TWO                                                                DATE……………….

TOPIC: GRAPHICAL METHOD OF SOLVING SIMULTANEOUS EQUATION

Expressions in x written as ax+b where a and b are constants (which can be any number) are known as linear expression. Thus we can draw a graph representing the above expression by equating it to y. to draw a linear graph we select suitable values of x so as to calculate the values of corresponding y. hence to draw a simultaneous equation, we make y the subject in each of the equation. Then find the values of the corresponding y with the selected suitable values of x.

Steps   in using graphical method

1. Make y the subject in each equation.
2. Draw a table of values for each of the linear equations; taking a range of values.
3. On a graph paper label the x-axis and y –axis according to the table drawn in step 2(two) above.
4. Plot these values and join the points for each of the table of values.
5. Take note of the point of intersection of the two lines. At this point trace it to both y and x axes. The values are the only pair of values that satisfy both simultaneous equations.

Example:

Solve graphical the simultaneous equation below

X + y =6 ; 3x –y = 12.

Solution:

1.       From eq(1)  y= 6- x

From eq(2) y= 3x+2

1. Draw the table of values of the equations taking a range of values ie

Table for y = 6-x

 X -1 0 1 2 3 y 7 6 5 4 3

Guiding equation: Y=6-x

1. when x= -1, Y =6-(-1), Y= 6 +1, Y=7
2. when x=0, Y= 6-0, Y= 6
3. when x=1, Y= 6-1, Y= 5
4. when x=2, Y= 6 -2, Y= 4
5. when x=3, Y=6-3, Y= 3

Table for Y=3x+2

Guiding equation: Y=3x+2

 X -1 0 1 2 3 y -1 2 5 8 11

1. when x=-1, Y=3(-1) + 2; Y= -3+2; Y= -1
2. when x= 0, Y= 3(0) + 2; Y= 0+2 = 2
3. when x=1, Y=5
4. when x=2, Y=8
5. when x=3, Y=3(3) +2; Y=9+2=11

EVALUATION

1. What is a linear equation?
2. Which of these equations are linear? A. a+b B.a2 +b =12 C. x-1 = 2
3. What is the first step in drawing graph?

Further exercises on the use of graph to solve simultaneous equation

In making of table of values for points to be plotted, x is called independent variable while y is the dependent variable. The point where the variable crosses an axis is called an intercept.

Example:

Draw the graph of the given pair of the equation 2x-y=3, x+y=6 and show the point of intercept of the lines on the y-axis.

Solution:

1. Make the y subject in each equation. I.e. Y=2x-3; Y=6-x
2. Make a table of values for each equation with ranges of Y=2x-3
 X -1 0 1 2 3 y -5 -3 -1 1 3
1. Make the table of value for Y=6-x
 X -1 0 1 2 3 y 7 6 5 4 3

From the graph the points of intercept are -3 and -6.

EVALUATION:

Solve graphically the below simultaneous equation:

1. Y-x = -4; Y+3x =12
2. 8c +3d = 1; 4c+5d =9

Essential mathematics for J.S.S 3 Pg 146-147

Exam focus for J.S.S CE Pg 218-219

WEEKEND ASSINMENT

1. How  many variables do we have in x+y+z-6=128 A.2 B. 1 C. 3
2. What axes are used in the plotting of graph? A. P&Q B. X&Y C.P&Y
3. when given that Y=2x-1, what is the y if x=-1 A. -3 B. -2 C. 1
4. Given that coordinates at the point of intercession of a drawn graph is (-1,3), the values of y is … A.-1 B. 3 C. 2
5. Make n the subject in 9m-4n= -36 n=      A. -36+9m        B.  9m+36       C.   36-9m

4                        4                      4

THEORY

Solve  the following pairs of simultaneous equations graphically:

1. 2x + y=8; x+y=5     2. 6x+y=12; x-y=9