1. Simultaneous equations
Sketching straight lines
Solving simultaneous equations by straight line graphs
Solving simultaneous equations by substitution
Solving simultaneous equations by elimination
Knowing two points which lie on a line find the equation of the line
Using simultaneous equations to solve problems
Simultaneous equation by substitution
Problems from credit past papers

2. Sketching straight lines
Equations of the type
y = x ; y = x + 3 ; y = 4x - 5; x + y = are equations of straight lines . The general equation of a straight line is
y = ax =b
By finding the coordinates of some points which lie on the straight lines, plotting them and joining them up the graph of the straight line can be drawn.

3. Table of values
To draw the graph of a straight line we must find the coordinates of some points which lie on the line.We do this by forming a table of values . Give the x coordinate a value and find the corresponding y coordinate for several points
Make a table of values for the equation y = 2x + 1 x
y=2x+1 1 2 3 3 7 1 5
So (0,1) (1,3) (2,5) and (3,7) all lie on the line with equation y=2x + 1
Now plot the points on a grid and join them up

4. . . . . y Plot the points (0,1) (1,3) (2,5) and (3,7) on the grid 7 64 3 2 1 .
Now join then up to give a straight line . . x
All the points on the line satisfy the equation y = 2x + 1

5. Sketching lines by finding where the lines cross the x axis and the y axis A quicker method
Straight lines cross the x axis when the value of y = o
Straight lines cross the y axis when the value of x =0
Sketch the line 2x + 3y = 6
Line crosses x axis when y = 0
Line crosses y axis when x = 0
2x + 0 =6
0 + 3y = 6
3y =6
2x =6
y = 2
x =3
at ( 3,0)
at ( 0,2)
Ex 2 page 124

6. Plot 0,2) and (3,0) and join them up with a straight liney 7 6 5 4 3 2 1
Now join then up to give a straight line . . x
All the points on the line satisfy the equation 2x + 3y = 6

7. Solving simultaneous equations by straight line graphs
Two lines either meet at a point intersect or they are parallel they never meet
To find where two lines meet draw the graphs of both lines on the same grid and read off the point of intersection

8. Find where the lines x + y = 5 and x – y =1 intersect
get two points that lie on each line or better still three
x + y = 5 1 5 x
(0,5) and (5,0) and (1,4)lie on the line x + y = 5 y 5 4
x – y =1 4 1 x
(0,-1) and (1,0) and ( 4,3) lie on the line y -1 3
Now plot the points and find where the two lines meet

9. . . . . . . x + y =1 (3,2) X x + y =5 The lines intersect at (3,2) y 76 5 4 3 2 1 -1 . .
x + y =1 .
(3,2) X
x + y =5 . . x .
The lines intersect at (3,2)

10. Solving simultaneous equations algebraically
Whoopee no more drawing graphs
But you will have to learn a strategy to solve the equations
Yikes!

11. Elimination method x + y = 8 x - y = 4 ADD
Solve the simultaneous equations algebraically
HOW DO I SOLVE TWO EQUATIONS WITH TWO UNKNOWNS ?
x + y = 8
x - y = 4
2x = 12
ADD the two equations together
This gives one equation with one unknown which is easy to solve
x = 6
ADD
How do I find y. It is gone.It has been eliminated ?

12. SUBSTITUTE THE VALUE OF X INTO ONE OF YOUR EQUATIONS
x + y = 8
Now I have the solution
6 + y = 8
y = 2
Am I correct?
Check by putting the values of x and y into , the other equation, x-y,and see if you get 4
6 – 2 = 
Solution is x =6 and y = 2

13. Solve the simultaneous equations
x + 3y = 7
x - 3y = -5
Remember to check
ADD
2x = 2
x = 1
Substitute x =1 into x + 3y = 7
y = 7
3y =6
1 – 3 X 2
y = 2
= 1 -6
Solution is x = 1 and y = 2
= 

14. Solve the simultaneous equations
If I add I get another equation in x and y
3x + y = 8
x + y = 4
3x + 2=12
no use
3x + y = 8
-x -y = -4
Multiply one of the equations by ( -1)
ADD
2x = 4
x = 2
It changes the sign of everything
SUBSTITUTE Put x=2 into x + y =4
x + y = y = 4
3X2 + 2 = 8 
y = 2
check
Solution x = 2 and y = 2

15. Using simultaneous equations to solve problems
example

16. Slightly more difficult
But not for you IF you have learned the work so far
Solve the simultaneous equations
Adding gives 5x + 5y = 55 no use
3x + 2y = 29
2x + 3y = 26
Multiplying by a negative does not eliminate any of the letters

17. Could I multiply and then add?
That ‘s good let’s try it but be careful
Remember when we add numbers together which are the negatives of each other they are eliminated

18. Solve the following simultaneous equations algebraically
Multiply the equations by two suitable numbers so that the coeficients of x or y are the negatives of each other
3x + 2y = 29 2x + 3y = 26
We choose y to be eliminated
X 3
3x + 2y = 29 2x + 3y = 26
X -2
Giving two new equivalent equations
9x +6y = 87
-4x + -6y = -52
Now add

19. Remember practice makes perfect
9x +6y = 87
-4x + -6y = -52
Remember practice makes perfect
5x = 35
ADD
x = 7
check
substitute
x = 7 into 9x+6y = 87
-4 X X 4
= -28 –24
= 
9 X y = 87
63 +6y = 87
6y = 24
y = 4
Ex 7B p 139
Solution x = 7 and y = 4

20. Using simultaneous equation to solve problems

21. Problem Solving With Simultaneous Equations
Example : The problem
For the cinema
2 adults’ tickets and 5 children’s tickets cost £26
4 adults’ tickets and 2 children tickets cost £28
Find the cost of each kind of ticket.
Introduce Letters
Let the cost of adult ticket = £A
Let the cost of children’s ticket = £C
Write the equations

22. Solve the Problem using simultaneous equations
Conclusion
Adult Ticket costs £5.50
Children's tickets cost £3

23. strategy Read the problem Introduce two letters
Write two equations that describe the information
Solve the problem using simultaneous equations
check

24. Substitution method
This an excellent method of solving simultaneous equations when the equations are given to you in a certain form
Solve the equations y = 2x and y = x + 10
Example 1
We take the equation y = x + 10
And substitute 2x for y
2x = x + 10
Take out y and
replace it with 2x
x = 5
Sub x=5 into y=2x
y = 10
Now solve the equation
Solution x = 5 y = 10

25. Example 2
Solve the equations y = 2x –8 and y – x = 1
Sub y =2x-8 in the equation
y x = 1
2x –8 -x = 1
x – 8 = 1
x = 9
y = 18-8
Sub x = 9 in the equation y = 2x -8
y = 10
Solution x = 9 and y =10
Check 10 –9 =1

26. Problems with simultaneous equations from past papers
1. The tickets for a sports club cost £2 for members and £3 for non- members
The total ticket money collected was £580
x tickets were sold to members and y tickets were sold to non-members.
Use this information to write down an equation involving x and y
2x + 3y =580

27. b) 250 people bought raffle tickets for the disco
b) people bought raffle tickets for the disco. Write down another equation involving x and y.
x + y = 250
c ) How many tickets were sold to members ?
We now have two equations in x and y so let’s solve them simultaneously
2x + 3y = 580
x y = 250

28. 2x + 3y = 580
x y = 250
X 1
X (-2)
2x y = 580
-2x + (-2 )y = -500
ADD
y = 80
Sub y = 80 into x + y = 250
check
x =250
2 x x80 = =580
x = 170
170 tickets were sold to members

29. 2. Alloys are made by mixing metals
2. Alloys are made by mixing metals.Two different alloys are made using iron and lead. To make the first alloy, 3 cubic cms of iron and 4 cubic cms of lead are used This alloy weighs grams
a ) Let x grams be the weight of 1 cm3 f iron and y grams be the weight of 1 cm3 of lead Write down an equation in x and y which satisfies the above equation.
3x + 4 y = 65
To make the second alloy, 5 cm3 of iron and 7cm3 of lead are used . This alloy weighs grams.
b ) Write down a second equation in x and y which satisfies this condition
5x + 7y = 112

30. C ) Find the weight of 1 cm3 of iron and the weight of 1cm3 of lead.
X (-5)
3x + 4 y = x + 7y = 112
X 3
Check x7 + 7x11
= = 112
-15x y = -325
15x y = 336
ADD
y = 11
y = 11 into 3x + 4y = 65
SUB
3x = 65
1cm3 of iron weighs 7gms and 1cm3 of lead weighs 11 gms
3x = 21
x = 7

31. 3. A rectangular window has length , l cm and breadth, b cm .
A security grid is made to fit this window. The grid as 5 horizontal wires and 8 vertical wires
a ) The perimeter of the window is 260 cm. Use this information to write down an equation involving l and b .
2 l + 2 b = 260
b) In total, 770 cm of wire is used. Write down another equation involving l and b
5 l + 8 b = 770

32. Find the length and breadth of the window
2 l + 2 b = 260
5 l + 8 b = 770
X (4)
X (-1)
8 l b =
-5 l + (-8) b =
ADD
3 l = l = 90
5x90 +8x40 = =770
l = 90 into 2 l + 2 b = 260
SUB
b = 260
Length is 90cm and the breadth is 40 cm
2b = 80
b = 40

33. 4. A number tower is built from bricks as shown in fig 1
4. A number tower is built from bricks as shown in fig 1. The number on the brick above is always equal to the sum of the two numbers below.
fig1 9 12 -3 8 4 -7 3 5 -1 -6

34. Find the number on the shaded brick in fig 234
Fig 2 16 18 5 11 7 34 -2 7 4
-11

35. -3 Fig 3 p q -5 2 p + 2q –5 + q -8 = -3 P + 3q – 13 = -3 simplifying
In fig 3, two of the numbers on the bricks are represented by p and q
Show that p + 3q = 10 -3
p +2q -5
q-8
p+q
q -5 -3
Fig 3 p q -5 2
Adding the numbers in the second row to equal the top row
p + 2q –5 + q -8 = -3
P + 3q – 13 = -3
simplifying
P + 3q = 10

36. d) Find the values of p and q
Use fig 4 to write down a second equation in p and q 14
2q+1
8-p
fig 4
2q-2 3
5-p 2q -2 5 -p
Adding the numbers in the second row to equal the top row
2q+1+8-p =14
2q +9 – p = 14
simplifying
2q – p = 5
d) Find the values of p and q

37. 3q + P = 10 2q – p = 5 3q + P = 10 2q - p = 5 ADD Check 3x3 +1 = 10
Change the terms around q under q and p under p
Remember to check
3q + P = 10
2q - p = 5
ADD
Check 3x3 +1 = 10
5q = 15
q = 3
SUB
q = 3 into 2q –p =5
p = 1 and q = 3
6 –p = 5
p=1

38. 5. A sequence of numbers is ,5 ,12 ,22 ,……… Numbers from this sequence can be illustrated in the following way using dots •
First number ( N = 1)
• •
• • •
Second number (N=2)
• • •
• •
• • • •
• •
• • •
Third number (N=3)
• •
• •
• • •
• •
• •
• •
• •
• • •
• •
Fourth number (N=4)

39. Find the values of a and b
a )Write the fifth number in the pattern
Form a table of values N 1 2 3 4 5
D= no of dots 12 22 35
When N= D = 5
When N= 1 D = 1
b) The number of dots needed to illustrate the nth number in this sequence is given by the formula
D =aN² - bN
Find the values of a and b

40. N = 1 D = When N= D = 5
D =aN² - bN
Form two equations by substituting the values of n and D into the equation D =aN² - bN
N = 1 D = 1
1 = a - b
5 = 4a -2b
N= D = 5
Now solve simultaneously

41. X (-2)
1 = a - b X1
5 = 4a -2b
-2 = -2a b
5 = 4a - 2b
Check
4x3/2-2x1/2
=12/2-2/2 =10/2=5
ADD
3 = 2a
a = 3/2
a = 3/3 b= 1/3
1 = 3/2 -b
SUB
b = 1/2