1. Simultaneous Equations
Algebra (2)
Simultaneous Equations

2. Line Equations Line graphs can be described using equations
This graph follows the equation y=x
Y is the y axis going up
X is the x axis going across
So if y=1 (up) then x has to be 1 (across)
If x is -1, what will y be? -1

3. Questions Which line would be steeper? Y=2x +1 Y=3x Y=½ x
Answer= Y= 3x

4. Drawing Graphs Draw a graph for Y= x + 1 Make a table for the points x4 3 2 1 x y 1 2 3 4 5 -4 -3 -2 -1 1 2 3 4 x -2 -3 -4

5. Gradients This is how steep the graph line is
In an equation the number before the x tells you the gradient
E.g.
3x +1 and 3x – 4 are parallel
The numbers afterwards tell you where the lines cross the y axis
+1 for the first line and -4 for the second

6. Questions Are these parallel 3x -2 and 2x -3 3x + 1 and 4x + 1
Y = 2x + 4
What is the equation for the red line?

7. Answers No
Yes
Y = 2x -2

8. Do these points lie on this line?
Does the point (2,5), lie on the line y = x + 4
You do not need to draw the graph, just workout the equation
Y = 2 + 4
Y = 6
This would be (2,6) not (2,5)

9. Questions Do these points lie on these lines? (3,7) on y = x + 4

10. Answers
Yes No

11. Drawing Graphs the Quick Way
Draw a graph for 2x + y = 6
Find out where x= 0 and y = 0 are
First x = 0
0 + 6 = 6 so (0,6)
Now y = 0
2x + 0 = 6
X= 3 (3,0)
2x + y = 6

12. Points of Intersection
Two lines cross at a point of intersection
Alex and Tom are buying some food at a youth club
Alex buys two biscuits and 1 drink for 10p
Tom buys one biscuit and two drinks for 14 p
Alex’s equation is 2b + d = 10
Tom’s is b + 2d= 14
How much is a biscuit and how much is a drink?

13. Example b The cost of 1 drink is 6p The cost of 1 biscuit is 2p d
Alex:
2b + d =10
When b= 0 d =10
(0,10)
When d= 0, b = 5
(5,0)
Tom:
b + 2d = 14
When b = 0, d = 7
(0,7)
When d = 0, 14
(14,0) 6p 2p b
The cost of 1 drink is 6p
The cost of 1 biscuit is 2p

14. Simultaneous Equations
When you solve two equations at the same time you have done a simultaneous equation
Example
Draw graphs to solve these equations
X + y = 5
2x + 4y = 12

15. Example Answer = (4,1) 2x + 4y = 12 X + y = 5 X + y = 5
If x= 0 then y = 5
(0,5)
If y = 0, then x = 5
(5,0)
2x + 4y = 12
If x= 0, then y = 3
(0,3)
If y = 0, then x = 6
(6,0)
Answer = (4,1)
2x + 4y = 12
X + y = 5

16. Solving Simultaneous Equations
Solve these simultaneous equations
5x + y = 13
X + y = 5
Get rid of the y by subtracting the two equations
4x = 8
X = 2
Now put x=2 into one of your equations the find out what y is
10 + y = 13
Y = 3
So x = 2, y = 3

17. Solving Simultaneous Equations
When subtracting does not work you have to add
2x + y = 8
X – y = 7
You have to add to get rid of the y
3x = 15
X = 5
No put this into one of your equations
10 + (-2) = 8
Check this with the other equation
5 – (-2) = 7
So x = 5 and y = -2

18. Solving Simultaneous Equations
Sometimes equations need to be multiplied first
3x + 2y = 16
X + y = 7
We want to get rid of the y’s so we need to multiply the second equation by 2
2x + 2y = 14
Now we can subtract
X = 2
Now we put this value into one of our equations
6 + 2y =16
2y = 10
Y = 5
So x = 2 and y = 5

19. Solving Simultaneous Equations
How would you solve
2x + 3y = 11
5x + 4y = 24
Multiply both equations
2x + 3y = 11 multiplied by 4
5x + 4y = 24 multiplied by 3
8x + 12y = 44
15x + 12y = 72
Now we can subtract
7x = 28
X = 4
Now place this value into our equation
8 + 3y = 11
3y = 3
Y = 1
So x = 4 and y = 1

20. Solving Simultaneous Equations
Solve these simultaneous equations
X = 2 + y
3x + y =14
We have to rearrange these equations so that the number is on its own
X – y = 2
How do we get rid of the y?
We add the two equations
4x = 16
X = 4
Then we put this value into one of our equations
4 – y = 2
Y = 2
So x = 4 and y = 2

21. Inequalities These lines all follow rules
This red line is always on x=7
This purple line is always on y = 3
This dark red line is always on
X= -10

22. Inequalities
Find the region where x ≤ 2

23. Inequalities Find the region where y < x + 2 This line is y= x + 2
Because y is less than, but not equal to this line it has to be drawn as a dotted line
This line is y=x
Less than means under the line

24. Inequalities Find the region where Y ≥ -3, x ≤ 2 and y < x x ≤ 2
This is the cross over region
Y ≥ -3