Algebraic Summary
MarkAnderson
Algebra-Simultaneousequations1.pptx
GCSE Mathematics – Simultaneous Equations
SLIDE NUMBER 1
May 2019
© VIDLEARN® 2019
Rebecca Wigfull
1
Session Objectives
The purpose of the session is to:
Solve simple simultaneous equations.
Solve complex simultaneous equations.
Solve simultaneous equations in context.
Solve linear and non linear simultaneous equations.
Solve simultaneous equations by finding approximate solutions using a graph.
SLIDE NUMBER 2
May 2019
© VIDLEARN® 2019
2
CONSIDER…
At this point you should consider the list of session objectives and ask yourself:
How many of the session objectives am I confident with
Could I explain these objectives in relation to teaching and learning
SLIDE NUMBER 3
May 2019
© VIDLEARN® 2019
3
Session Objectives
The purpose of the session is to:
Solve simple simultaneous equations.
Solve complex simultaneous equations.
Solve simultaneous equations in context.
Solve linear and non linear simultaneous equations.
Solve simultaneous equations by finding approximate solutions using a graph.
SLIDE NUMBER 4
May 2019
© VIDLEARN® 2019
4
Simple simultaneous equations
Simultaneous Equations
SLIDE NUMBER 5
May 2019
© VIDLEARN® 2019
2x - y = 1
2x + 2y = 10
1
2
2x - y = 1
2x + 2y = 10
-3y = -9
y = 3
-
Sub y = 3 into
1
2x - 1x3 = 1
2x - 3 = 1
2x = 4
x = 2
Sub y = 3 and x = 2 into
2
2x2 + 2x3 = 10
4 + 6 = 10
10 = 10
x = 2 and y = 3
(2,3)
Check
Same Sign Subtract (SSS)
5
Simple simultaneous equations
Simultaneous Equations
SLIDE NUMBER 6
May 2019
© VIDLEARN® 2019
2x - 5y = 1
3x + 5y = 14
1
2
2x - 5y = 1
3x + 5y = 14
5x = 15
x = 3
+
Sub x = 3 into
1
2x3 - 5y = 1
6 - 5y = 1
- 5y = -5
y = 1
Sub x = 3 and y = 1 into
2
3x3 + 5x1 = 14
9 + 5 = 14
14 = 14
x = 3 and y = 1
(3,1)
Check
6
Review of main ideas from above:
Solve the following simultaneous equations
SLIDE NUMBER 7
May 2019
© VIDLEARN® 2019
CONSIDER…
a)
3x - y = 18
3x + 6y = -3
b)
3x - 2y = 57
5x - 2y = 111
c)
2x - y = 7
4x + y = 23
x = 5 and y = -3
(5,-3)
x = 27 and y = 12
(27,12)
x = 5 and y = 3
(5,3)
7
Review of main ideas from above:
Solutions
SLIDE NUMBER 8
May 2019
© VIDLEARN® 2019
CONSIDER…
a)
3x – y = 18
3x + 6y = -3
1
2
3x - y = 18
3x + 6y = -3
-7y = 21
y = -3
-
Sub y = -3 into
1
3x - 1x-3 = 18
3x + 3 = 18
3x = 15
x = 5
Sub x = 5 and y = -3 into
2
3x5 + 6x-3 = -3
15 - 18 = -3
-3 = -3
x = 5 and y = -3
(5,-3)
Check
Same Sign Subtract (SSS)
8
Review of main ideas from above:
Solutions
SLIDE NUMBER 9
May 2019
© VIDLEARN® 2019
CONSIDER…
b)
3x - 2y = 57
5x - 2y = 111
1
2
3x - 2y = 57
5x - 2y = 111
-2x =-54
x = 27
-
Sub x = 27 into
1
3x27 - 2y = 57
81 - 2y = 57
- 2y =-24
y = 12
Sub x = 27 and y = 12 into
2
5x27 - 2x12 = 111
135 - 24 = 111
111 = 111
x = 27 and y = 12
(27,12)
Check
Same Sign Subtract (SSS)
9
Review of main ideas from above:
Solutions
SLIDE NUMBER 10
May 2019
© VIDLEARN® 2019
CONSIDER…
1
2
2x - y = 7
4x + y = 23
6x = 30
x = 5
+
Sub x = 5 into
1
2x5 - y = 7
10 - y = 7
- y = -3
y = 3
Sub x = 5 and y = 3 into
2
4x5 + 1x3 = 23
20 + 3 = 23
23 = 23
x = 5 and y = 3
(5,3)
Check
c)
2x - y = 7
4x + y = 23
10
Session Objectives
The purpose of the session is to:
Solve simple simultaneous equations.
Solve complex simultaneous equations.
Solve simultaneous equations in context.
Solve linear and non linear simultaneous equations.
Solve simultaneous equations by finding approximate solutions using a graph.
SLIDE NUMBER 11
May 2019
© VIDLEARN® 2019
11
Engineering a match
Simultaneous Equations
SLIDE NUMBER 12
May 2019
© VIDLEARN® 2019
2x + 3y = 4
x - 2y = -5
1
2
2x + 3y = 4
2x - 4y = -10
7y = 14
y = 2
-
Sub y = 2 into
1
2x + 3x2 = 4
2x + 6 = 4
2x = -2
x = -1
Sub x = -1 and y = 2 into
2
1x-1 - 2x2 = -5
-1 - 4 = -5
-5 = -5
x = -1 and y = 2
(-1,2)
Check
Same Sign Subtract (SSS)
Multiply by 2
2x - 4y = -10
2
3
12
Engineering a match
Simultaneous Equations
SLIDE NUMBER 13
May 2019
© VIDLEARN® 2019
3x + 5y = 7
4x + 3y = 2
1
2
9x + 15y = 21
20x + 15y = 10
-11x = 11
x = - 1
-
Sub x = -1 into
1
3x-1 + 5y = 7
-3 + 5y = 7
5y = 10
y = 2
Sub x = -1 and y = 2 into
2
4x-1 + 3x2 = 2
-4 + 6 = 2
2 = 2
x = -1 and y = 2
(-1,2)
Check
Same Sign Subtract (SSS)
Multiply by 3
9x + 15y = 21
1
3
Multiply by 5
20x + 15y = 10
2
4
13
Review of main ideas from above:
Solve the following simultaneous equations
SLIDE NUMBER 14
May 2019
© VIDLEARN® 2019
CONSIDER…
a)
2x + 3y = 8
5x - 2y = 1
b)
3x - y = 18
x + 2y = -1
c)
2x - 5y = 3
3x + 2y = 14
x = 1 and y = 2
(1,2)
x = 5 and y = -3
(5,-3)
x = 4 and y = 1
(4,1)
14
Review of main ideas from above:
Solutions
SLIDE NUMBER 15
May 2019
© VIDLEARN® 2019
CONSIDER…
a)
2x + 3y = 8
5x - 2y = 1
x = 1 and y = 2
(1,2)
1
2
4x + 6y = 16
15x - 6y = 3
19x = 19
x = 1
+
Sub x = 1 into
1
2x1 + 3y = 8
2 + 3y = 8
3y = 6
y = 2
Sub x = 1 and y = 2 into
2
5x1 - 2x2 = 1
5 - 4 = 1
1 = 1
Check
Multiply by 2
4x + 6y = 16
1
3
Multiply by 3
15x - 6y = 3
2
4
15
SLIDE NUMBER 16
May 2019
© VIDLEARN® 2019
CONSIDER…
b)
3x - y = 18
x + 2y = -1
x = 5 and y = -3
(5,-3)
Review of main ideas from above:
Solutions
1
2
3x - y = 18
3x + 6y = -3
-7y = 21
y = -3
-
Sub y = -3 into
1
3x - 1x-3 = 18
3x + 3 = 18
3x = 15
x = 5
Sub x = 5 and y = -3 into
2
1x5 + 2x-3 = -1
5 - 6 = -1
-1 = -1
Check
Multiply by 3
3x + 6y = -3
2
3
Same Sign Subtract (SSS)
16
SLIDE NUMBER 17
May 2019
© VIDLEARN® 2019
CONSIDER…
c)
2x - 5y = 3
3x + 2y = 14
x = 4 and y = 1
(4,1)
Review of main ideas from above:
Solutions
1
2
6x - 15y = 9
6x + 4y = 28
-19y = -19
y = 1
-
Sub y = 1 into
1
2x - 5x1 = 3
2x - 5 = 3
2x = 8
x = 4
Sub x = 4 and y = 1 into
2
3x4 + 2x1 = 14
12 + 2 = 14
14 = 14
Check
Multiply by 3
6x - 15y = 9
1
3
Multiply by 2
6x + 4y = 28
2
4
Same Sign Subtract (SSS)
17
Session Objectives
The purpose of the session is to:
Solve simple simultaneous equations.
Solve complex simultaneous equations.
Solve simultaneous equations in context.
Solve linear and non linear simultaneous equations.
Solve simultaneous equations by finding approximate solutions using a graph.
SLIDE NUMBER 18
May 2019
© VIDLEARN® 2019
18
Questions in context – writing equations
Simultaneous Equations
SLIDE NUMBER 19
May 2019
© VIDLEARN® 2019
3 men and 2 women earn £460 per week at a company. 2 men and 1 woman earn £280. Assuming all men earn the same amount, as do the women, how much does each man and woman earn?
3m
3 men and 2 women earn £460 per week at a company. 2 men and 1 woman earn £280. Assuming all men earn the same amount, as do the women, how much does each man and woman earn?
3m + 2w
3 men and 2 women earn £460 per week at a company. 2 men and 1 woman earn £280. Assuming all men earn the same amount, as do the women, how much does each man and woman earn?
3m +
3 men and 2 women earn £460 per week at a company. 2 men and 1 woman earn £280. Assuming all men earn the same amount, as do the women, how much does each man and woman earn?
3m + 2w = 460
3 men and 2 women earn £460 per week at a company. 2 men and 1 woman earn £280. Assuming all men earn the same amount, as do the women, how much does each man and woman earn?
2m
3 men and 2 women earn £460 per week at a company. 2 men and 1 woman earn £280. Assuming all men earn the same amount, as do the women, how much does each man and woman earn?
2m +
3 men and 2 women earn £460 per week at a company. 2 men and 1 woman earn £280. Assuming all men earn the same amount, as do the women, how much does each man and woman earn?
3 men and 2 women earn £460 per week at a company. 2 men and 1 woman earn £280. Assuming all men earn the same amount, as do the women, how much does each man and woman earn?
2m + w = 280
2m + w
3 men and 2 women earn £460 per week at a company. 2 men and 1 woman earn £280. Assuming all men earn the same amount, as do the women, how much does each man and woman earn?
19
Questions in context - solving
Simultaneous Equations
SLIDE NUMBER 20
May 2019
© VIDLEARN® 2019
3m + 2w = 460 2m + w = 280
1
2
3m + 2w = 460
4m + 2w = 560
-m = -100
m = 100
-
Sub m = 100 into
1
3x100 + 2w = 460
300 + 2w = 460
2w = 160
w = 80
Sub m = 100 and w = 80 into
2
2x100 + 1x80 = 280
200 + 80 = 280
280 = 280
Men earn £100 Women earn £80
Check
Same Sign Subtract (SSS)
Multiply by 2
4m + 2w = 560
2
3
20
Review of main ideas from above:
Solve the following simultaneous equations
SLIDE NUMBER 21
May 2019
© VIDLEARN® 2019
CONSIDER…
3 lots of a number ‘x’ added to 2 lots of a number ‘y’ comes to 12. Also, 4 lots of the same number ‘x’ minus 1 lots of the number ‘y’ comes to 5. What are the values of ‘x’ and ‘y’?
3 fish minus 2 bags of chips cost £6. Also, 5 fish and 6 lots of chips costs £38. How much does one fish and a bag of chips cost?
x = 2 and y = 3
Twice one number added to three times another is 21. The difference between the number is three. Find the numbers.
Fish = £4 Chips = £3 Total £7
3 and 6
21
Review of main ideas from above:
Solutions
SLIDE NUMBER 22
May 2019
© VIDLEARN® 2019
CONSIDER…
3x + 2y = 12
4x - y = 5
x = 2 and y = 3
1
2
3x + 2y = 12
8x - 2y = 10
11x = 22
x = 2
+
Sub x = 2 into
1
3x2 + 2y = 12
6 + 2y = 12
2y = 6
y = 3
Sub x = 2 and y = 3 into
2
4x2 - 1x3 = 5
8 - 3 = 5
5 = 5
Check
Multiply by 2
8x - 2y = 10
2
3
3 lots of a number ‘x’ added to 2 lots of a number ‘y’ comes to 12. Also, 4 lots of the same number ‘x’ minus 1 lots of the number ‘y’ comes to 5. What are the values of ‘x’ and ‘y’?
22
Review of main ideas from above:
Solutions
SLIDE NUMBER 23
May 2019
© VIDLEARN® 2019
CONSIDER…
3x - 2y = 6
5x + 6y = 38
1
2
9x - 6y = 18
5x + 6y = 38
14x = 56
x = 4
+
Sub x = 4 into
1
3x4 - 2y = 6
12 - 2y = 6
- 2y = -6
y = 3
Sub x = 4 and y = 3 into
2
5x4 + 6x3 = 38
20 + 18 = 38
38 = 38
Check
Multiply by 3
9x - 6y = 18
1
3
3 fish minus 2 bags of chips costs £6. Also, 5 fish and 6 lots of chips costs £38. How much does one fish and a bag of chips cost?
Fish = £4 Chips = £3 Total £7
23
Review of main ideas from above:
Solutions
SLIDE NUMBER 24
May 2019
© VIDLEARN® 2019
CONSIDER…
2x + 3y = 21
x - y = 3
1
2
2x + 3y = 21
3x - 3y = 9
5x = 30
x = 6
+
Sub x = 6 into
1
2x6 + 3y = 21
12 + 3y = 21
3y = 9
y = 3
Sub x = 6 and y = 3 into
2
1x6 – 1x3 = 3
6 – 3 = 3
3 = 3
Check
Multiply by 3
3x - 3y = 9
2
3
Twice one number added to three times another is 21. The difference between the number is three. Find the numbers.
6 and 3
24
Session Objectives
The purpose of the session is to:
Solve simple simultaneous equations.
Solve complex simultaneous equations.
Solve simultaneous equations in context.
Solve linear and non linear simultaneous equations.
Solve simultaneous equations by finding approximate solutions using a graph.
SLIDE NUMBER 25
May 2019
© VIDLEARN® 2019
25
Simultaneous Equations
SLIDE NUMBER 26
May 2019
© VIDLEARN® 2019
y = x2 + 2x – 4
y = x + 2
x2 + 2x – 4 = x + 2
x2 + x – 6 = 0
(x+3)(x-2) = 0
x = -3 or 2
x = -3
y = x + 2
y = -3 + 2
y = -1
(-3,-1)
x = 2
y = x + 2
y = 2 + 2
y = 4
(2,4)
-1 = -32 + 2x-3 – 4
-1 = 9 -6 -4
-1 = -1
4 = 22 + 2x2 – 4
4 = 4 + 4 – 4
4 = 4
Check
Linear and non-linear simultaneous equations
26
Simultaneous Equations
SLIDE NUMBER 27
May 2019
© VIDLEARN® 2019
y = x2 – 3
y – 2x = 5
x2 – 3 = 2x + 5
x2 – 2x – 8 = 0
(x+2)(x-4) = 0
x = -2 or 4
x = -2
y = 2x + 5
y = -4 + 5
y = 1
(-2,1)
x = 4
y = 2x4 + 5
y = 8 + 5
y = 13
(4,13)
1 = -22 – 3
1 = 4 – 3
1 = 1
13 = 42 – 3
13 = 16 – 3
13 = 13
Check
y = 2x + 5
Linear and non-linear simultaneous equations
27
Simultaneous Equations
SLIDE NUMBER 28
May 2019
© VIDLEARN® 2019
x2 + y2 = 1
2x + y = 1
x2 + (1-2x)(1-2x) = 1
5x2 - 4x = 0
x = 0 or 0.8
x = 0
y = 1 – 2x
y = 1
(0,1)
x = 0.8
y = 1 – 2x
y = 1 – 1.6
y = -0.6
(0.8,-0.6)
02 + 12 = 1
1 = 1
0.82 + -0.62 = 1
0.64 + 0.36 = 1
1 = 1
Check
y = 1 – 2x
x(5x – 4) = 0
x2 + 1 - 4x + 4x2 = 1
Linear and non-linear simultaneous equations
28
Linear and non-linear simultaneous equations
Simultaneous Equations
SLIDE NUMBER 29
May 2019
© VIDLEARN® 2019
3x2 + 2xy – 6 = 0
2y – 7x = 2
3x2 + 2x(1+3.5x) – 6 = 0
10x2 + 2x – 6 = 0
y = 1 + 3.5x
(0.68,3.38)
y = 1 + 3.5x
y = -2.083587387
y = -2.08
(-0.88,-2.08)
y = 1 + 3.5x
3x2 +2x + 7x2 – 6 = 0
Factorise using quadratic formula
x = -1 ± √61
10
y = 3.383587387
x = -1 + √61
10
x = 0.68 (2dp)
x = -1 - √61
10
x = -0.88 (2dp)
x = -1 + √61
10
x = -1 - √61
10
y = 3.38 (2dp)
29
Review of main ideas from above:
Solve the following simultaneous equations, round answers to 2dp is needed.
SLIDE NUMBER 30
May 2019
© VIDLEARN® 2019
CONSIDER…
a)
y= 2x2 - 3x + 4
y = 4x + 1
b)
3x + 2y = 7
y= x2 – 2x + 3
c)
x + y = 3
x2 + y2 = 25
(0.5, 3) and
(3 , 13)
(-0.5,4.25) and
(1 , 2)
(4.70,-1.70) and
(-1.70, 4.70)
30
Review of main ideas from above:
Solutions
SLIDE NUMBER 31
May 2019
© VIDLEARN® 2019
CONSIDER…
a)
y= 2x2 - 3x + 4
y = 4x + 1
(0.5, 3) and
(3 , 13)
2x2 - 3x + 4 = 4x + 1
2x2 - 7x + 3 = 0
(2x-1)(x-3) = 0
x = 0.5 or 3
x = 0.5
y = 4x0.5 + 1
y = 2 + 1
y = 3
(0.5,3)
x = 3
y = 4x3 + 1
y = 12 + 1
y = 13
(3,13)
3 = 2x0.52 -3x0.5 + 4
3 = 0.5 - 1.5 + 4
3 = 3
13 = 2x32 - 3x3 + 4
13 = 18 - 9 + 4
13 = 13
Check
31
Review of main ideas from above:
Solutions
SLIDE NUMBER 32
May 2019
© VIDLEARN® 2019
CONSIDER…
x2 - 2x + 3 = 3.5 - 1.5x
x2 - 0.5x - 0.5 = 0
(x-1)(x+0.5) = 0
x = -0.5 or 1
x = -0.5
y = 3.5-1.5x-0.5
y = 3.5 + 0.75
y = 4.25
(-0.5,4.25)
x = 1
y = 3.5 - 1.5x1
y = 3.5 - 1.5
y = 2
(1,2)
4.25 = -0.52 -2x-0.5 + 3
4.25 = 0.25 + 1 + 3
4.25 = 4.25
2 = 12 - 2x1 + 3
2 = 1 - 2 + 3
2 = 2
Check
b)
3x + 2y = 7
y= x2 – 2x + 3
(-0.5,4.25) and (1 , 2)
y = 3.5 - 1.5x
y= x2 – 2x + 3
32
Review of main ideas from above:
Solutions
SLIDE NUMBER 33
May 2019
© VIDLEARN® 2019
CONSIDER…
c)
x + y = 3
x2 + y2 = 25
y = 3 - x
x2 + y2 = 25
(4.70,-1.70) and
(-1.70, 4.70)
x2 + (3-x)(3-x) = 25
2x2 - 6x - 16 = 0
y = 3 - x
(4.70,-1.70)
y = 3 - x
y = 4.701562119
y = 4.70
(-1.70,4.70)
x2 + 9 - 6x + x2 = 25
Factorise using quadratic formula
x = 3 ± √41
2
y = -1.701562119
x = 3 + √41
2
x = 4.70 (2dp)
x = 3 - √41
2
x = -1.70 (2dp)
y = -1.70 (2dp)
x2 - 3x - 8 = 0
x = 3 + √41
2
x = 3 - √41
2
33
Session Objectives
The purpose of the session is to:
Solve simple simultaneous equations.
Solve complex simultaneous equations.
Solve simultaneous equations in context.
Solve linear and non-linear simultaneous equations.
Solve simultaneous equations by finding approximate solutions using a graph.
SLIDE NUMBER 34
May 2019
© VIDLEARN® 2019
34
Solving simultaneous equations graphically
Simultaneous Equations
SLIDE NUMBER 35
May 2019
© VIDLEARN® 2019
x + 3y = 9
2x + y = 8
x + 3y = 9
When
x = 0
3y = 9
y = 3
When
y = 0
x = 9
2x + y = 8
When
x = 0
y = 8
x = 4
When
y = 0
2x = 8
x
x
x
x
(3,2)
35
Solving simultaneous equations graphically
Simultaneous Equations
SLIDE NUMBER 36
May 2019
© VIDLEARN® 2019
y = x + 1
Y = 2x + 3 – x2
(-1,0)
y = x + 1
When
x = 0
y = 1
When
y = 0
x = -1
and (2,3)
| x | y | x | y |
| -5 | -32 | 1 | 4 |
| -4 | -21 | 2 | 3 |
| -3 | -12 | 3 | 0 |
| -2 | -5 | 4 | -5 |
| -1 | 0 | 5 | -12 |
| 0 | 3 |
36
Solving simultaneous equations graphically
Simultaneous Equations
SLIDE NUMBER 37
May 2019
© VIDLEARN® 2019
y = 4x + 2
x2 + y2 = 36
(1,5.9)
y = 4x + 2
When
x = 0
y = 2
When
y = 0
4x = -2
and
(-1.9,-5.7)
x = -0.5
37
Review of main ideas from above:
Solve the following simultaneous equations, graphically.
SLIDE NUMBER 38
May 2019
© VIDLEARN® 2019
CONSIDER…
a)
y= x2 - 2x -3
2y = x + 1
b)
y = 2x -3
y = 4 - 0.5x
c)
3x + 2y = 6
x2 + y2 = 4
(-1,0) and
(3.5,2.25)
(2.8,2.6)
(0.8,1.8) and
(2,0)
38
y = 0.5
Review of main ideas from above:
Solutions
SLIDE NUMBER 39
May 2019
© VIDLEARN® 2019
CONSIDER…
a)
y= x2 - 2x -3
2y = x + 1
(-1,0)
2y = x + 1
When
x = 0
2y = 1
When
y = 0
x = -1
and (3.5,2.25)
| x | y | x | y |
| -5 | 32 | 1 | -4 |
| -4 | 21 | 2 | -3 |
| -3 | 12 | 3 | 0 |
| -2 | 5 | 4 | 5 |
| -1 | 0 | 5 | 12 |
| 0 | -3 |
39
Review of main ideas from above:
Solutions
SLIDE NUMBER 40
May 2019
© VIDLEARN® 2019
CONSIDER…
b)
y = 2x -3
y = 4 - 0.5x
y = 2x-3
When
x = 0
y = -3
x = 1.5
When
y = 0
2x = 3
y = 4 - 0.5x
When
x = 0
y = 4
x = 8
When
y = 0
-0.5x = -4
(2.8,2.6)
x
x
x
x
40
Review of main ideas from above:
Solutions
SLIDE NUMBER 41
May 2019
© VIDLEARN® 2019
CONSIDER…
c)
3x + 2y = 6
x2 + y2 = 4
(0.8,1.8)
3x + 2y = 6
When
x = 0
2y = 6
When
y = 0
3x = 6
and (2,0)
x = 2
y = 3
41
Session Objectives
The purpose of the session is to:
Solve simple simultaneous equations.
Solve complex simultaneous equations.
Solve simultaneous equations in context.
Solve linear and non-linear simultaneous equations.
Solve simultaneous equations by finding approximate solutions using a graph.
SLIDE NUMBER 42
May 2019
© VIDLEARN® 2019
42
SLIDE NUMBER 43
May 2019
© VIDLEARN® 2019
CONSIDER…
End of Presentation
At this point it would be advisable to go back over the presentation. Ensure that you are fully able to deal accurately and effectively with each session objective.
You should supplement the content of this session with suitable reading, research and discussion with others.
End of presentation
Rebecca Wigfull
SLIDE NUMBER 44
May 2019
© VIDLEARN® 2019
GCSE Mathematics – Simultaneous Equations
44
